How to factor polynomials: Difference between revisions

Revision as of 09:14, 16 May 2010

Revision as of 02:18, 17 May 2010

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The methods that are used to factor a polynomial depend on how many terms the polynomial has.<br /> Note: This page assumes that the polynomials are being factored on the real field (that the answers are real numbers).

==Any Polynomialpolynomial==

==Any Polynomial==

The first step to factor '''any''' polynomial is '''always''' to factor out the '''Greatest Common Factor''', commonly referred to as the '''GCF'''.

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: <math>4x(x+2)+3x^2(x+2)=(x+2)(4x+3x^2)</math>

==Binomial (2two Termsterms)==

Again, the first step is to factor out the GCF. If there is no GCF, then there are only 3three possibilities:

'''Difference of Squaressquares''', '''Sumsum of Cubescubes''', or '''Differencedifference of Cubescubes'''.

===Difference of Squaressquares===

<math>x^2-y^2=(x+y)(x-y)</math>

'''For example:'''<br />

: <math>y^2-9=(y+3)(y-3),</math><br />

<br />or

or

<math>16a^2-49b^2=(4a+7b)(4a-7b).</math>

: <math>16a^2-49b^2=(4a+7b)(4a-7b).</math>

===Sum of Cubes===

<math>x^3+y^3=(x+y)(x^2-xy+y^2)</math>

===Sum of Cubescubes===

'''For example:'''<br />

<math>z^3+27=(z+3)(z^2-3z+9),</math><br />

: <math>x^3+y^3=(x+y)(x^2-xy+y^2)</math>

<br />or

'''For example:'''<br />

: <math>z^3+27=(z+3)(z^2-3z+9),</math><br />

or

: <math>8x^3+125=(2x)^3+(5)^3=(2x+5)[(2x)^2-(5)(2x)+(5)^2]=(2x+5)(4x^2-10x+25).</math>

===Difference of Cubescubes===

<math>x^3-y^3=(x-y)(x^2+xy+y^2)</math>

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<math>8x^3-125=(2x)^3-(5)^3=(2x-5)[(2x)^2+(5)(2x)+(5)^2]=(2x-5)(4x^2+10x+25).</math>

==Trinomial (3three Termsterms)==

There are three possibilities for factoring a trinomial depending on which type of trinomial it is.

===Monic Trinomialstrinomials===

A monic trinomial has 1 as the leading coefficient.<br />

<math>x^2+bx+c=(x+d)(x+e),</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

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<math>x^2-11x+24=(x-3)(x-8)</math> because <math>(-3)(-8)=24</math> and <math>-3+-8=-11</math>

===Non-Monicmonic Trinomialstrinomials===

A non-monic trinomial has a constant other than 1 as the leading coefficient.

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Therefore <math>6x^2+7x-3=(3x-1)(2x+3)</math>

===Perfect Squaresquare Trinomialstrinomials===

Perfect square trinomials are of the form <math>a^2+2ab+b^2</math> or <math>a^2-2ab+b^2</math><br />

<math>a^2+2ab+b^2=(a+b)^2</math> and<br />

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<math>100x^2+180xy+81y^2=(10x)^2+2(10x)(9y)+(9y)^2=(10x+9y)^2</math>

==Polynomials with 4four terms==

PolynomialsSome polynomials with 4four terms can be factored by some form of grouping. There are special groupings but the most common form is referred to as '''factoring by grouping''' and is described step by step below.

'''Example:''' <br /><math>a^2-3ab+4ac-12bc</math>