{"id":2688,"date":"2023-03-23T14:43:18","date_gmt":"2023-03-23T09:13:18","guid":{"rendered":"https:\/\/www.tutoroot.com\/blog\/?p=2688"},"modified":"2026-03-06T12:34:24","modified_gmt":"2026-03-06T07:04:24","slug":"what-are-polynomials-definition-types-examples","status":"publish","type":"post","link":"https:\/\/www.tutoroot.com\/blog\/what-are-polynomials-definition-types-examples\/","title":{"rendered":"What are Polynomials? Definition, Types, Examples"},"content":{"rendered":"<p><span class=\"NormalTextRun SCXW186491655 BCX0\">Polynomials are algebraic expressions with variables and coefficients in them.<\/span> <span class=\"NormalTextRun SCXW186491655 BCX0\">Polynomials are a type of mathematical dialect. They are used to express numbers in <\/span><span class=\"NormalTextRun SCXW186491655 BCX0\">almost every<\/span><span class=\"NormalTextRun SCXW186491655 BCX0\"> field of mathematics and play <\/span><span class=\"NormalTextRun SCXW186491655 BCX0\">an essential role<\/span><span class=\"NormalTextRun SCXW186491655 BCX0\"> in others, such as calculus.<\/span><\/p>\n<p><span class=\"TextRun SCXW186491655 BCX0\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW186491655 BCX0\">Polynomials include \\(2x+9\\), \\(x^{2}+3x+11\\). <span class=\"TextRun SCXW207091505 BCX0\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW207091505 BCX0\">You may have <\/span><span class=\"NormalTextRun SCXW207091505 BCX0\">observed<\/span><span class=\"NormalTextRun SCXW207091505 BCX0\"> that none of these examples uses the &#8220;=&#8221; symbol. Check out this article to have a better understanding of polynomials.<\/span><\/span><span class=\"EOP SCXW207091505 BCX0\" data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/span><\/span><\/p>\n<h2><b><span data-contrast=\"auto\">What is a Polynomial?<\/span><\/b><\/h2>\n<p><span data-contrast=\"auto\">The word polynomial originated from the Greek &#8216;poly&#8217;, which means &#8216;many&#8217;, and &#8216;nominal&#8217; means &#8216;terms&#8217; therefore, altogether, it is described as &#8220;many terms&#8221;.<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><span data-contrast=\"auto\">A polynomial is a mathematical expression that does not contain the equal to sign (=). Let us look at the definitions and examples of polynomials given below.<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<h3><b><span data-contrast=\"auto\">Definition of Polynomial<\/span><\/b><\/h3>\n<p><span data-contrast=\"auto\">A polynomial is a sort of algebraic statement in which all variable exponents are whole numbers. Variable exponents in any polynomial must be non-negative integers. A polynomial is made up of constants and variables, where we cannot divide polynomials by variables.<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><iframe loading=\"lazy\" title=\"Polynomials - Definition, Degree, and Types | Part 1 | Grade 10 Maths \ud83d\udcd0\u2728 #polynomials  #mathbasics\" width=\"1170\" height=\"658\" src=\"https:\/\/www.youtube.com\/embed\/DyhFate14PM?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/p>\n<h2><b><span data-contrast=\"auto\">Notation and Standard Form of Polynomial<\/span><\/b><\/h2>\n<p><strong>\\(p(x)= a_{n} x^{n}+a_{n-1} x^{n-1}+a_{n-2} x^{n-2}+&#8230;&#8230;+ a_{1}x+ a_{0}\\)<\/strong><\/p>\n<p>Where, \\(a_{n}, a_{n-1}, a_{n-2},&#8230;.\\) <span class=\"TextRun SCXW85070639 BCX0\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW85070639 BCX0\">are termed coefficients of \\(x^{n}, x^{n-1}, x^{n-2},&#8230;.\\) <span class=\"TextRun SCXW24227786 BCX0\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW24227786 BCX0\">and constant term, correspondingly, and it should be a real number <\/span><span class=\"NormalTextRun SCXW24227786 BCX0\">(\u22f2<\/span><span class=\"NormalTextRun SCXW24227786 BCX0\"> R).<\/span><\/span><span class=\"EOP SCXW24227786 BCX0\" data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/span><\/span><\/p>\n<h2><b><span data-contrast=\"auto\">Types of Polynomials<\/span><\/b><\/h2>\n<p><span data-contrast=\"auto\">Polynomials are classified into the following groups based on the number of terms:<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<ul>\n<li><span data-contrast=\"auto\">Monomial<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/li>\n<li><span data-contrast=\"auto\">Binomial<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/li>\n<li><span data-contrast=\"auto\">Trinomial<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/li>\n<\/ul>\n<p><span data-contrast=\"auto\">A polynomial with four terms (Quadronomial)<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><span data-contrast=\"auto\">Polynomial with 5 terms (pentanomial), and so on&#8230;<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><span data-contrast=\"auto\">These polynomials can be coupled with addition, subtraction, multiplication, and division, but never with a variable.<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<h4 aria-level=\"3\"><b><span data-contrast=\"none\">Monomial<\/span><\/b><\/h4>\n<p><span data-contrast=\"none\">A monomial expression has only one term. The single term in an expression must be non-zero for it to be a monomial. These are some examples of monomials:<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p>\\(5x\\)<br \/>\n\\(3\\)<br \/>\n\\(6 a^{4}\\)<br \/>\n\\(-3xy\\)<\/p>\n<h4 aria-level=\"3\"><b><span data-contrast=\"none\">Binomial<\/span><\/b><\/h4>\n<p><span data-contrast=\"none\">A binomial expression has two terms. A binomial is the sum or difference of two or more monomials. These are some examples of binomials:<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p>\\(-5x+3\\)<br \/>\n\\(6 a^{4}+17x\\)<br \/>\n\\(x y^{2}+xy\\)<\/p>\n<h4 aria-level=\"3\"><b><span data-contrast=\"none\">Trinomial<\/span><\/b><\/h4>\n<p><span data-contrast=\"none\">A trinomial expression has three terms. Below are some examples of trinomial expressions:<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p>\\(-8 a^{3}+4x+7\\)<br \/>\n\\(4 x^{2}+9x+7\\)<\/p>\n<h3><b><span data-contrast=\"auto\">Examples of Polynomials<\/span><\/b><\/h3>\n<p><span data-contrast=\"auto\">Let me illustrate this using an example: 3&#215;2 + 5. We need to understand the given terms. In this situation, the variable is x. Where product 3 multiplied by x2 has a different name. The term &#8220;coefficient&#8221; is used to describe it. The number 5 is known as the constant. The variable x has a power of two.<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<h2><strong><span class=\"TextRun SCXW156030323 BCX0\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW156030323 BCX0\">Polynomial Examples<\/span><\/span><\/strong><\/h2>\n<h4><strong>Polynomial Addition Examples<\/strong><\/h4>\n<p><strong>Ex. Add \\(x^{2}-x+5\\) and \\(6 x^{2}+2x-10\\)<\/strong><\/p>\n<p><strong>Ans.\u00a0<\/strong> As given, Let&#8217;s add the above equations. Then,<\/p>\n\\((x^{2}-x+5)+(6 x^{2}+2x-10)\\)\n<p>Then<\/p>\n\\(=&gt;x^{2}-x+5+6 x^{2}+2x-10\\)\n<p>Grouping Like terms,<\/p>\n\\((x^{2}+6 x^{2})+(-x+2x)+(5-10)\\)\n<p><strong>\\(=&gt;7 x^{2}+x-5\\)<\/strong><\/p>\n<h4><strong>Polynomial Subtraction Examples<\/strong><\/h4>\n<p><strong>Ex. Subtract \\(2 x^{2}-6x+12\\) from \\(3 x^{2}-8x+7\\)<\/strong><\/p>\n<p><strong>Ans.\u00a0<\/strong>As given, Let&#8217;s Subtract the above equations.<\/p>\n\\(3 x^{2}-8x+7-(2 x^{2}-6x+12)\\)\n<p>Then<\/p>\n\\(3 x^{2}-8x+7-2 x^{2}+6x-12\\)\n<p>Grouping the Like Terms,<\/p>\n\\((3 x^{2}-2x^{2})+(-8x+6x)+(7-12)\\)\n<p><strong>\\(x^{2}-2x-5\\)<\/strong><\/p>\n<h2><b><span data-contrast=\"none\">Properties of Polynomials<\/span><\/b><\/h2>\n<p><span data-contrast=\"none\">Several key polynomial characteristics, as well as some notable polynomial theorems, are as follows:<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><b><span data-contrast=\"none\">Property 1:<\/span><\/b><span data-contrast=\"none\"> Division Algorithm<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><span data-contrast=\"none\">When a polynomial P(x) is divided by a polynomial G(x), the quotient Q(x) with the remainder R(x) is obtained.<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><span data-contrast=\"none\">P(x) = G(x) \u2022 Q(x) + R(x)<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><span data-contrast=\"none\">Where R(x)=0 or the degree of R(x) &lt; the degree of G(x)<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><b><span data-contrast=\"auto\">Property 2:<\/span><\/b><span data-contrast=\"auto\"> Bezout\u2019s Theorem<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><span data-contrast=\"auto\">Polynomial P(x) is divisible by binomial (x \u2013 a) if and only if P(a) = 0.<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><b><span data-contrast=\"auto\">Property 3:<\/span><\/b><span data-contrast=\"auto\"> Remainder Theorem<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><span data-contrast=\"auto\">If P(x) is divided by (x \u2013 a) with remainder r, then P(a) = r.<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><b><span data-contrast=\"auto\">Property 4:<\/span><\/b><span data-contrast=\"auto\"> Factor Theorem<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><span data-contrast=\"auto\">A polynomial P(x) divided by Q(x) results in R(x) with zero remainders if and only if Q(x) is a factor of P(x).<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><b><span data-contrast=\"auto\">Property 5:<\/span><\/b><span data-contrast=\"auto\"> Intermediate Value Theorem<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><span data-contrast=\"auto\">If P(x) is a polynomial, and P(x) \u2260 P(y) for (x &lt; y), then P(x) takes every value from P(x) to P(y) in the closed interval [x, y].<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><b><span data-contrast=\"auto\">Property 6<\/span><\/b><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><span data-contrast=\"auto\">The addition, subtraction, and multiplication of polynomials P and Q are<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><span data-contrast=\"auto\">Degree (P \u00b1 Q) \u2264 Degree (P or Q)<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><span data-contrast=\"auto\">Degree (P \u00d7 Q) = Degree(P) + Degree(Q)<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335551550&quot;:1,&quot;335551620&quot;:1,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><b><span data-contrast=\"auto\">Property 7<\/span><\/b><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><span data-contrast=\"auto\">Every zero of Q is likewise a zero of P if a polynomial P is divisible by a polynomial Q.<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><b><span data-contrast=\"auto\">Property 8<\/span><\/b><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><span data-contrast=\"auto\">P is divisible by (Q \u2022 R) if it is divisible by two co-prime polynomials Q and R.<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><strong><span class=\"TextRun SCXW59008286 BCX0\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW59008286 BCX0\">Property 9<\/span><\/span><\/strong><\/p>\n<p>If, \\(p(x)= a_{0}+ a_{1}x+ a_{2} x^{2}+&#8230;..+ a_{n} x^{n}\\) <span class=\"TextRun SCXW184662268 BCX0\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW184662268 BCX0\">is a polynomial such that deg(P) = n \u2265 0 then, P has at most \u201cn\u201d distinct roots.<\/span><\/span><\/p>\n<p><b><span data-contrast=\"auto\">Property 10:<\/span><\/b><span data-contrast=\"auto\"> Descartes&#8217; Sign Rule<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><span data-contrast=\"auto\">The number of positive real zeroes in a polynomial function P(x) is equal to or less than the number of changes in the sign of the coefficients by an even integer. If the &#8220;K&#8221; sign changes, the number of roots is &#8220;k&#8221; or &#8220;(k &#8211; a)&#8221;, where &#8220;a&#8221; is an even integer.<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p aria-level=\"3\"><b><span data-contrast=\"none\">Property 11: <\/span><\/b><span data-contrast=\"none\">Fundamental Theorem of Algebra<\/span><span data-ccp-props=\"{&quot;134245418&quot;:true,&quot;134245529&quot;:true,&quot;201341983&quot;:0,&quot;335559738&quot;:40,&quot;335559739&quot;:0,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><span data-contrast=\"none\">There is at least one complex zero in every non-constant single-variable polynomial with complex coefficients.<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p aria-level=\"3\"><b><span data-contrast=\"none\">Property 12<\/span><\/b><span data-ccp-props=\"{&quot;134245418&quot;:true,&quot;134245529&quot;:true,&quot;201341983&quot;:0,&quot;335559738&quot;:40,&quot;335559739&quot;:0,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p aria-level=\"3\"><span class=\"TextRun SCXW262459489 BCX0\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"none\"><span class=\"NormalTextRun SCXW262459489 BCX0\">f P(x) is a real-coefficient polynomial with one complex zero (x = a &#8211; bi), then x = a + bi is also a zero of P. (x). <\/span><\/span><\/p>\n<p aria-level=\"3\">Therefore, \\(x^{2}-2ax+ a^{2}+ b^{2}\\) <span class=\"TextRun SCXW36406383 BCX0\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"none\"><span class=\"NormalTextRun SCXW36406383 BCX0\">will be a P factor (x).<\/span><\/span><\/p>\n<h2><b><span data-contrast=\"none\">Polynomial Operations<\/span><\/b><\/h2>\n<p><span data-contrast=\"none\">Basic algebraic procedures can be applied to polynomials of various forms. These four fundamental polynomial operations are as follows:<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<ul>\n<li><span data-contrast=\"auto\">Addition of polynomials<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/li>\n<li><span data-contrast=\"auto\">Subtraction<\/span><span data-contrast=\"auto\"> of Polynomial<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/li>\n<li><span data-contrast=\"auto\">Polynomial multiplication<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/li>\n<li><span data-contrast=\"auto\">Dividing polynomials<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/li>\n<\/ul>\n<h4 aria-level=\"3\"><b><span data-contrast=\"none\">Addition of Polynomials<\/span><\/b><\/h4>\n<p><span data-contrast=\"none\">Adding polynomials always requires similar terms that are with the same variable and power. In the Polynomial addition, the result is the same polynomial degree.<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<h4 aria-level=\"3\"><b><span data-contrast=\"none\">Subtraction of Polynomials<\/span><\/b><\/h4>\n<p><span data-contrast=\"none\">Subtracting polynomials is the same as addition, the only difference is the style of operation. Where you have to subtract the like terms. Also, note that the subtraction of polynomials results in the same degree polynomial.<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<h4 aria-level=\"3\"><b><span data-contrast=\"none\">Multiplication of Polynomials<\/span><\/b><\/h4>\n<p><span data-contrast=\"none\">When two or more polynomials are multiplied, they always result in a higher degree polynomial.<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<h4 aria-level=\"3\"><b><span data-contrast=\"none\">Division of Polynomials<\/span><\/b><\/h4>\n<p><span data-contrast=\"none\">A polynomial can be formed by dividing two polynomials. Let&#8217;s take a closer look at polynomial division. Follow these steps to divide polynomials:<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<h4><b><span data-contrast=\"none\">Polynomial Division Steps<\/span><\/b><\/h4>\n<ul>\n<li><span data-contrast=\"none\">We utilize the long division approach when a polynomial contains more than one term. The steps are as follows.<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/li>\n<li><span data-contrast=\"none\">In decreasing sequence, write the polynomial.<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/li>\n<li><span data-contrast=\"none\">Examine the greatest power and divide the phrases by it.<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/li>\n<li><span data-contrast=\"none\">As the division symbol, use the solution from step 2.<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/li>\n<li><span data-contrast=\"none\">Subtract it from the following word and bring it down.<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/li>\n<li><span data-contrast=\"none\">Steps 2\u20134 should be repeated until there are no more words to carry down.<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/li>\n<li><span data-contrast=\"none\">It is important to note that the final answer, including the remainder, will be in fraction form (last subtract term).<\/span><span data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/li>\n<\/ul>\n<h2><b><span data-contrast=\"auto\">Examples of Polynomial Operations<\/span><\/b><\/h2>\n<h4><strong><span class=\"TextRun SCXW111759796 BCX0\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW111759796 BCX0\">Addition of Polynomials<\/span><\/span><\/strong><\/h4>\n<p><strong>Ex. \\((2 x^{2}+3x+2)+(3 x^{2}-5x-1)\\)<\/strong><\/p>\n<p><strong>Ans.\u00a0<\/strong>\\(2 x^{2}+3x+2+3 x^{2}-5x-1\\)<\/p>\n<p>Adding like terms,<\/p>\n\\((2 x^{2}+3 x^{2})+(3x-5x)+(2-1)\\)\n<p>Finally,<\/p>\n\\(5 x^{2}-2x+1\\)\n<h4><strong><span class=\"TextRun SCXW111759796 BCX0\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW111759796 BCX0\">Multiplication of Polynomials<\/span><\/span><\/strong><\/h4>\n<p><strong>Ex. \\((6x\u22123y) \u00d7 (2x+5y)\\)<\/strong><\/p>\n<p><strong>Ans.<\/strong> As Given,<\/p>\n\\(6x \u00d7(2x+5y)\u20133y \u00d7 (2x+5y)\\)\n<p>Therefore,<\/p>\n\\((12 x^{2}+30xy)-(6yx+15 y^{2})\\)\n<p>Then,<\/p>\n\\(12 x^{2}+24xy-15 y^{2} \\)\n<p><a class=\"maths-track-poly\" href=\"https:\/\/www.tutoroot.com\/maths-online-tuition?ref=polynomials_c10_img\" target=\"_blank\" rel=\"noopener\"><img loading=\"lazy\" class=\"alignnone wp-image-6308 size-large\" src=\"https:\/\/www.tutoroot.com\/blog\/wp-content\/uploads\/2023\/03\/Polynomials-CTA-1024x576.png\" alt=\"Polynomials Class 10 Formulas and Zeroes of Polynomial Chart\" width=\"1024\" height=\"576\" srcset=\"https:\/\/www.tutoroot.com\/blog\/wp-content\/uploads\/2023\/03\/Polynomials-CTA-1024x576.png 1024w, https:\/\/www.tutoroot.com\/blog\/wp-content\/uploads\/2023\/03\/Polynomials-CTA-300x169.png 300w, https:\/\/www.tutoroot.com\/blog\/wp-content\/uploads\/2023\/03\/Polynomials-CTA-768x432.png 768w, https:\/\/www.tutoroot.com\/blog\/wp-content\/uploads\/2023\/03\/Polynomials-CTA-1536x864.png 1536w, https:\/\/www.tutoroot.com\/blog\/wp-content\/uploads\/2023\/03\/Polynomials-CTA-2048x1152.png 2048w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/a><\/p>\n<h2><strong>Final Notes<\/strong><\/h2>\n<p><span class=\"TextRun SCXW186045521 BCX0\" lang=\"EN-US\" xml:lang=\"EN-US\" data-contrast=\"auto\"><span class=\"NormalTextRun SCXW186045521 BCX0\">Enroll in <\/span><span class=\"NormalTextRun SCXW186045521 BCX0\">Tutoroot<\/span><span class=\"NormalTextRun SCXW186045521 BCX0\"> to receive <\/span><span class=\"NormalTextRun SCXW186045521 BCX0\"><a href=\"https:\/\/www.tutoroot.com\/maths-online-tuition\"><strong><span class=\"ui-provider a b c d e f g h i j k l m n o p q r s t u v w x y z ab ac ae af ag ah ai aj ak\" dir=\"ltr\">one on one maths online tuition classes<\/span><\/strong><\/a> on various topics. <\/span><span class=\"NormalTextRun SCXW186045521 BCX0\">Register today to have access to a plethora of courses covering various subjects more effectively.<\/span><\/span><span class=\"EOP SCXW186045521 BCX0\" data-ccp-props=\"{&quot;201341983&quot;:0,&quot;335559739&quot;:160,&quot;335559740&quot;:259}\">\u00a0<\/span><\/p>\n<p><span data-teams=\"true\">For more simplified explanations like the one above, visit the maths blogs on the Tutoroot website. Elevate your learning with Tutoroot\u2019s personalised Maths online tuition. Begin your journey with a FREE DEMO session and discover the advantages of <a href=\"https:\/\/www.tutoroot.com\/\"><strong>one on one Online Tuitions<\/strong><\/a>.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Polynomials are algebraic expressions with variables and coefficients in them. Polynomials are a type of mathematical dialect. They are used to express numbers in almost every field of mathematics and &hellip; <a href=\"https:\/\/www.tutoroot.com\/blog\/what-are-polynomials-definition-types-examples\/\" class=\"more-link\">Read More<\/a><\/p>\n","protected":false},"author":7,"featured_media":6307,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[15],"tags":[53,30,29,110],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v19.4 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What are Polynomials? Definition, Types, Examples<\/title>\n<meta name=\"description\" content=\"Visit here to learn polynomials along with the types, notations, and examples. 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