Factoring Polynomials

Useful Formulas

Computing factors of polynomials requires knowledge of different formulas and some experience to find out which formula to be applied. Below, we give some important formulas:

${x^{2}-y^{2}=(x+y)(x-y)}$

${x^{2}+2xy+y^{2}=(x+y)^{2}}$

${x^{2}-2xy+y^{2}=(x-y)^{2}}$

${x^{3}-y^{3}=(x-y)(x^{2}+xy+y^{2})}$

${x^{3}+y^{3}=(x+y)(x^{2}-xy+y^{2})}$

${a^{3}+b^{3}+c^{3}-3abc=(a+b+c)(a^{2}+b^{2}+c^{2}-ab-ac-bc)}$

Methods

Given the expression $x^{2}+3x+2$ , one may ask "what are the values of $x$ that make this expression 0?" If we factor we obtain

$x^{2}+3x+2=(x+2)(x+1)$

.

If $x=-1,-2$ , then one of the factors on the right becomes zero. Therefore, the whole must be zero. So, by factoring we have discovered the values of $x$ that render the expression zero. These values are termed "roots." In general, given a quadratic polynomial $px^{2}+qx+r$ that factors as

$px^{2}+qx+r=(ax+c)(bx+d)$

then we have that $x=-{\frac {c}{a}}$ and $x=-{\frac {d}{b}}$ are roots of the original polynomial.

A special case to be on the look out for is the difference of two squares, $a^{2}-b^{2}$ . In this case, we are always able to factor as

$a^{2}-b^{2}=(a+b)(a-b)$

For example, consider $4x^{2}-9$ . On initial inspection we would see that both $4x^{2}$ and $9$ are squares of $2x$ and $3$, respectively. Applying the previous rule we have

$4x^{2}-9=(2x+3)(2x-3)$

The AC method

There is a way of simplifying the process of factoring using the AC method. Suppose that a quadratic polynomial has a formula of

$ax^{2}+bx+c$

If there are numbers $r$ and $q$ that satisfy both

$r\cdot q=ac$ and $r+q=b$

Then, we can rewrite the polynomial as

$ax^{2}+bx+c=ax^{2}+rx+qx+c$

and factor out a common term from $(ax^{2}+rx)$ and $(qx+c)$ to factor the polynomial.

For example, take the polynomial $2x^{2}+7x-15$.

$ac=2(-15)=-30$. $10(-3)=-30=ac$ and $10+(-3)=7=b$, so we can rewrite the polynomial as $2x^{2}+10x-3x-15$. From here, we can factor $2x$ from the first two terms, and $-3$ from the last two, which gives us $2x(x+5)+(-3)(x+5)=(2x-3)(x+5)$. Thus, the factored form of $2x^{2}+7x-15$ is $(2x-3)(x+5)$. Note that if we reverse the order of $10x$ and $-3x$, we get $2x^{2}-3x+10x-15=x(2x-3)+5(2x-3)=(x+5)(2x-3)$, which is the same factored form that we got previously. Thus, the order of $rx$ and $qx$ should not matter when using the AC method to factor a quadratic polynomial.

The quadratic formula

Given any quadratic equation $ax^{2}+bx+c=0\ ,\ a\neq 0$, all solutions of the equation are given by the quadratic formula:

$x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}$

Note that the value of $b^{2}-4ac$ will affect the number of real solutions of the equation.

If	Then
$b^{2}-4ac>0$	There are two real solutions for the equation
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^2-4ac=0}	There are only one real solutions for the equation
$b^{2}-4ac<0$	There are no real solutions for the equation

Remainder and Factor Theorem

The polynomial division algorithm is as follows: suppose $d(x)$ and $p(x)$ are nonzero polynomials where the degree of $p(x)$ is greater than or equal to the degree of $d(x)$. Then there exist two unique polynomials, $q(x)$ and $r(x)$, such that $p(x)=d(x)q(x)+r(x)$, where either $r(x)=0$ or the degree of $r(x)$ is strictly less than the degree of $d(x)$.

Remainder Theorem

Suppose $p(x)$ is a polynomial of degree at least 1 and c is a real number. When $p(x)$ is divided by $(x-c)$ the remainder is $p(c)$.

Proof: By the division algorithm, $p(x)=(x-c)q(x)+r$, where r must be a constant since $d(x)=x-c$ has a degree of 1. $p(x)=(x-c)q(x)+r$ must hold for all values of $x$, so we can set $x=c$ and get that $p(c)=(c-c)q(x)+r=r$. Thus the remainder $r=p(c)$.

Factor Theorem

Suppose $p(x)$ is a nonzero polynomial. The real number $c$ is a zero of $p(x)$ if and only if $(x-c)$ is a factor of $p(x)$.

By the division algorithm, $x-c$ is a factor of $p(x)$ if and only if $r=0$. So, since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(c) = r } when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) } is divided by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x - c } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x - c } is a factor of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) } if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(c) = 0 } ; that is, if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c } is a zero of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) } .

Factor Theorem Example Problem

Determine if x + 2 is a factor of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2x^2 + 3x -2} .

Since c is positive instead of negative we need to use this basic identity:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x + 2 = x - \left ( - 2 \right )}

Now we can use the factor theorem.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \left (-2 \right )^2 + 3 \left (-2 \right ) -2 = 8 - 6 - 2 = 0} .

Since the resultant is 0, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x+2)} is a factor of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2x^2 + 3x -2} .

This means it is possible to re-state the polynomial in the form (x+2)( some linear expression of x).

So Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2x^2 + 3x -2 = (x+2)(ax+b)}

Expanding the right hand side we get :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2x^2 + 3x -2 = ax^2 + x( 2a+b) +2b}

Equating like terms we get :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2= a}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2a+b = 3 } , and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2b = -2 }

Giving Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = 2} , $b=-1$ from the first and third equations and this works in the second, so

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2x^2 + 3x -2 = (x+2)(2x-1)}

Example Problems

EXAMPLE 1: Find all the roots of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4x^2+7x-2}

Finding the roots is equivalent to solving the equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4x^2+7x-2=0} . Applying the quadratic formula with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=4\ ,\ b=7\ ,\ c=-2} , we have:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\frac{-7\pm\sqrt{7^2-4(4)(-2)}}{2(4)}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\frac{-7\pm\sqrt{49+32}}{8}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\frac{-7\pm\sqrt{81}}{8}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\frac{-7\pm9}{8}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\frac{2}{8}\ ,\ x=\frac{-16}{8}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\frac{1}{4}\ ,\ x=-2}

The quadratic formula can also help with factoring, as the next example demonstrates.

EXAMPLE 2: Factor the polynomial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4x^2+7x-2}

We already know from the previous example that the polynomial has roots Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\frac{1}{4}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=-2} . Our factorization will take the form
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C(x+2)\left(x-\tfrac{1}{4}\right)}
.

All we have to do is set this expression equal to our polynomial and solve for the unknown constant C:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C(x+2)\left(x-\tfrac{1}{4}\right)=4x^2+7x-2}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C\left(x^2+\left(-\tfrac{1}{4}+2\right)x-\tfrac{2}{4}\right)=4x^2+7x-2}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C\left(x^2+\tfrac{7}{4}x-\tfrac{1}{2}\right)=4x^2+7x-2}

You can see that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C=4} solves the equation. So the factorization is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4x^2+7x-2=4(x+2)\left(x-\tfrac{1}{4}\right)=(x+2)(4x-1)}

EXAMPLE 3: Factor the polynomial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6x^2 + 7x + 2 } .

For this problem, let's use the AC method. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = 6 } , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = 7 } , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c = 2 } . So, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ac = 6(2) = 12 } , and we need to find two numbers whose product equals 12, and whose sum equals 7. Both ac and b are positive, so we are only concerned with the positive factors of 12, which are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1, 2, 3, 4, 6, } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 12 } . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3(4) = 12 = ac } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 + 4 = 7 = b } , so we can rewrite the polynomial as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6x^2 + 3x + 4x + 2 = (3x)(2x + 1) = 2(2x + 1) = (3x + 2)(2x+1)} .

Resources

• Factoring Polynomials with Example Problems, Paul's Online Notes
• Factoring Polynomials, Lumen Learning
• Difference of Squares. Produced by TA Catherine Sporer, UTSA
• 6 Methods of Factoring with Practive Problems, AlgebraLAB
• AC Factoring Method, Texas Wesleyan University

Licensing

Content obtained and/or adapted from:

• Algebra, Wikibooks: Calculus under a CC BY-SA license
• Factoring Polynomials, Wikibooks: Algebra under a CC BY-SA license