{"id":283,"date":"2017-06-02T13:43:47","date_gmt":"2017-06-02T08:13:47","guid":{"rendered":"http:\/\/ibelitetutor.com\/blog\/?p=283"},"modified":"2023-08-14T12:49:24","modified_gmt":"2023-08-14T07:19:24","slug":"how-to-factor-a-polynomial","status":"publish","type":"post","link":"http:\/\/ibelitetutor.com\/blog\/how-to-factor-a-polynomial\/","title":{"rendered":"How To Factorize a Polynomial"},"content":{"rendered":"

How to factor a polynomial<\/span><\/h2>\n

According to our IB Maths Tutors<\/a>,<\/strong> first of all, we need to understand the meaning of factorization. Factorization means writing a higher degree polynomial as a product of linear polynomials.<\/p>\n

Suppose we have a quadratic polynomial and we want to factorize it then we have to try to write it as a product of two linear polynomials.<\/p>\n

If we have a cubic polynomial and we want to factorize it then we have to\u00a0try to write it as a product of three linear polynomials. This process continues for all higher degree polynomials.<\/p>\n

Types of polynomial<\/span><\/h3>\n

There are many different types of polynomials classified on the basis of their degree and their number of terms, we have\u00a0a different way of factorization for almost every type of polynomials<\/p>\n

Factorization of a Monomial- Monomial is already a linear polynomial with degree one so we don’t need to factorize it.<\/p>\n

how to factor a polynomial with two terms<\/span><\/h3>\n

A polynomial with two\u00a0terms is called a binomial. we can have binomials of many types<\/p>\n

Binomial of degree two <\/span><\/h4>\n

When both terms have the same signs-<\/strong> these types of polynomials can’t be factorized, only a few can be factorized using perfect square identities.<\/p>\n

When both terms have opposite signs and the power of the variable is divisible by two-<\/strong><\/p>\n

these polynomials can easily be factorized by using a\u00b2-b\u00b2=(a+b)(a-b) identity<\/p>\n

Example: 9x\u00b2-16y\u00b2<\/strong><\/p>\n

=(3x)\u00b2-(4y)\u00b2<\/p>\n

= (3x-4y)(3x+4y)<\/p>\n

we can also factorize polynomials for degree\u00a04, degree 6, and degree 8<\/strong> \u00a0and much more in the same way<\/p>\n

When both terms have opposite signs and the power of the variable is divisible by three- these<\/strong> polynomials can easily be factorized by using a\u00b3-b\u00b3=(a-b)<\/strong>(a\u00b2+ab+b\u00b2) or a\u00b3+b\u00b3=(a+b)(a\u00b2-ab+b\u00b2)\u00a0identity<\/p>\n

Example: 64x\u00b3-27y\u00b3<\/strong><\/p>\n

\u00a0<\/strong> \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 =(4x)\u00b3-(3y)\u00b3<\/p>\n

= (3x-4y)(9x\u00b2+12xy+16y\u00b2)<\/p>\n

we can also factorize polynomials for degree 6, and degree\u00a09<\/strong> and much more in the same way<\/p>\n

How to factor a polynomial with three terms<\/span><\/h3>\n

A polynomial with three terms is called a cubic polynomial. A trinomial is usually a quadratic trinomial. This can be of two types:<\/p>\n

A perfect square quadratic trinomial can be solved using this identity<\/span><\/h4>\n

(a+b)\u00b2=a\u00b2+2ab+b\u00b2 or by (a-b)\u00b2=a\u00b2-2ab+b\u00b2<\/p>\n

Example- 9x\u00b2-24x+16<\/strong><\/p>\n

=(3x)\u00b2-2(3x)(4)+(4)\u00b2<\/p>\n

=(3x-4)\u00b2<\/p>\n

A generic (non-perfect square) quadratic trinomial then we factorize it using the middle term splitting method.<\/span><\/h4>\n
\n

Example: 9x\u00b2-25x+16<\/strong><\/p>\n

=9x\u00b2-(16x+9x)+16<\/p>\n

=9x\u00b2-16x-9x+16<\/p>\n

=x(9x-16)-1(9x-16)<\/p>\n

=(9x-1)96x-1)<\/p>\n

Factorization of cubic polynomials with four terms-these polynomials can be factorized in different ways.<\/strong><\/p>\n

Factorization by using hit and trial method-<\/span><\/h4>\n

we use this method for cubic polynomials of 3 or 4 terms when we have only one variable in the polynomials. hit and trial is used when terms are usually in order<\/p>\n

\n
Example:<\/em><\/h5>\n

Find the zeros of f(x<\/em>) = 2x3<\/sup> + 3x<\/em>2<\/sup> \u2013 11x<\/em> \u2013 6<\/strong><\/p>\n

Solution:<\/strong><\/em><\/h5>\n

We will find one solution to this polynomial by hit and trial method<\/p>\n

Step 1: Use the factor to test the possible values by hit and trial.<\/p>\n

f(1) = 2 + 3 \u2013 11 \u2013 6 \u2260 0<\/p>\n

f(\u20131) = \u20132 + 3 + 11 \u2013 6 \u2260 0<\/p>\n

f(2) = 16 + 12 \u2013 22 \u2013 6 = 0<\/p>\n

We find that the integer root is 2.<\/p>\n

Step 2: Find the other roots either by inspection or by synthetic division. I am showing the inspection method here, you should try division method yourself<\/p>\n

2x3<\/sup> + 3x<\/em>2<\/sup> \u2013 11x<\/em> \u2013 6
\n= (x<\/em> \u2013 2)(ax<\/em>2<\/sup> + bx + c<\/em>)
\n= (x<\/em> \u2013 2)(2x<\/em>2<\/sup> + bx + <\/em>3)
\n= (x<\/em> \u2013 2)(2x<\/em>2<\/sup> + 7x + <\/em>3)
\n= (x<\/em> \u2013 2)(2x<\/em> + 1)(x<\/em> +3)<\/p>\n

we have calculated a b and c by inspection or comparison method<\/p>\n

We can use binomial whole cube identity to factorize cubic polynomials that are perfect cubes in itself.<\/strong><\/span><\/h4>\n

(a+b)\u00b3=a\u00b3+3a\u00b2b+3ab\u00b2+b\u00b3<\/strong><\/span><\/p>\n

We can use this method to factorize cubic polynomials with four terms also but generally, we use it for 2 variables when two terms are perfect cubes and rest two are divisible by 3<\/p>\n

Example:\u00a0<\/strong>27x\u00b3+108x\u00b2y+144xy\u00b3+64y\u00b3<\/strong><\/p>\n

=(3x)\u00b3+3(3x)\u00b2(4y)+3(3x)(4y)+(4y)\u00b3<\/p>\n

=(3x+4y)\u00b3<\/p>\n

Besides these methods we can use :<\/span><\/h4>\n

We can also use this method to factorize cubic polynomials with 4 terms. generally, we use it for 2 or 3 variables when 3 terms are perfect cubes and 4th term is divisible by 3<\/p>\n

a\u00b3+b\u00b3+c\u00b3-3abc=(a+b+c)(a\u00b2+b\u00b2+c\u00b2-ab -bc -ca)<\/strong><\/p>\n

<\/h5>\n

If we are ever asked to evaluate or factorize a\u00b3+b\u00b3+c\u00b3 we should first find the sum of a+b+c usually this sum is zero then we can use<\/strong><\/p>\n

a\u00b3+b\u00b3+c\u00b3=3abcd<\/h5>\n

click red text to download questions .pdf\u00a0Factoring_Polynomials (1)<\/p>\n

\"ib<\/p>\n

For free Class Whatsapp at +919911262206 or fill the form<\/strong><\/h5>\n\n
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     <\/p>\n","protected":false},"excerpt":{"rendered":"

    How to factor a polynomial According to our IB Maths Tutors, first of all, we need to understand the meaning of factorization. Factorization means writing […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11],"tags":[],"class_list":["post-283","post","type-post","status-publish","format-standard","hentry","category-cbse-tutors"],"_links":{"self":[{"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/posts\/283","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/comments?post=283"}],"version-history":[{"count":0,"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/posts\/283\/revisions"}],"wp:attachment":[{"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/media?parent=283"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/categories?post=283"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/ibelitetutor.com\/blog\/wp-json\/wp\/v2\/tags?post=283"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}