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Factoring Completely

Sum or difference with higher powers, factoring algorithm, and grouping.

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Factoring Completely

Factoring Completely 

We say that a polynomial is factored completely when we can’t factor it any more. Here are some suggestions that you should follow to make sure that you factor completely:

  • Factor all common monomials first.
  • Identify special products such as difference of squares or the square of a binomial. Factor according to their formulas.
  • If there are no special products, factor using the methods we learned in the previous sections.
  • Look at each factor and see if any of these can be factored further.

Factoring Completely - Learn by Example

1. 6x230x+36

Factor out the common monomial. In this case 6 can be divided from each term:

6(x25x+6)

There are no special products. We factor x25x+6 as a product of two binomials: (x )(x )

The two numbers that multiply to 6 and add to -5 are -2 and -3, so:

6(x25x+6)=6(x2)(x3)

If we look at each factor we see that we can factor no more.

The answer is 6(x2)(x3).

2. 2x28

Factor out common monomials: 2x28=2(x24)

We recognize x24 as a difference of squares. We factor it as (x+2)(x2).

If we look at each factor we see that we can factor no more.

The answer is 2(x+2)(x2).

3. x3+6x2+9x

Factor out common monomials: x3+6x2+9x=x(x2+6x+9)

We recognize x2+6x+9 as a perfect square and factor it as (x+3)2.

If we look at each factor we see that we can factor no more.

The answer is x(x+3)2.

4. 2x4+162

Factor out the common monomial. In this case, factor out -2 rather than 2. (It’s always easier to factor out the negative number so that the highest degree term is positive.)

2x4+162=2(x481)

We recognize expression in parenthesis as a difference of squares. We factor and get:

2(x29)(x2+9)

If we look at each factor we see that the first parenthesis is a difference of squares. We factor and get:

2(x+3)(x3)(x2+9)

If we look at each factor now we see that we can factor no more.

The answer is 2(x+3)(x3)(x2+9).

5. x58x3+16x

Factor out the common monomial: x58x3+14x=x(x48x2+16)

We recognize x48x2+16 as a perfect square and we factor it as x(x24)2.

We look at each term and recognize that the term in parentheses is a difference of squares.

We factor it and get ((x+2)(x2))2, which we can rewrite as (x+2)2(x2)2.

If we look at each factor now we see that we can factor no more.

The final answer is x(x+2)2(x2)2.

Factor out a Common Binomial

The first step in the factoring process is often factoring out the common monomials from a polynomial. But sometimes polynomials have common terms that are binomials. For example, consider the following expression:

x(3x+2)5(3x+2)

Since the term (3x+2) appears in both terms of the polynomial, we can factor it out. We write that term in front of a set of parentheses containing the terms that are left over:

(3x+2)(x5)

This expression is now completely factored.

Factoring out Common Binomials - Learn by Example

1. 3x(x1)+4(x1)

3x(x1)+4(x1) has a common binomial of (x1).

When we factor out the common binomial we get (x1)(3x+4).

2. x(4x+5)+(4x+5)

x(4x+5)+(4x+5) has a common binomial of (4x+5).

When we factor out the common binomial we get (4x+5)(x+1).

Example

Factor completely: 24x328x2+8x.

First, notice that each term has 4x as a factor. Start by factoring out 4x:

24x328x2+8x=4x(6x27x+2)

Next, factor the trinomial in the parenthesis. Since a1 find ac: 62=12. Find the factors of 12 that add up to -7. Since 12 is positive and -7 is negative, the two factors should be negative:

121212=112=26=34 and  and  and 1+12=132+6=83+4=7

Rewrite the trinomial using 7x=3x4x, and then factor by grouping:

6x27x+2=6x23x4x+2=3x(2x1)2(2x1)=(3x2)(2x1)

The final factored answer is:

4x(3x2)(2x1)

Review

Factor completely.

  1. \begin{align*}2x^2+16x+30\end{align*}
  2. \begin{align*}5x^2-70x+245\end{align*}
  3. \begin{align*}-x^3+17x^2-70x\end{align*}
  4. \begin{align*}2x^4-512\end{align*}
  5. \begin{align*}25x^4-20x^3+4x^2\end{align*}
  6. \begin{align*}12x^3+12x^2+3x\end{align*}
  7. \begin{align*}12c^2-75\end{align*}
  8. \begin{align*}6x^2-600\end{align*}
  9. \begin{align*}-5t^2-20t-20\end{align*}
  10. \begin{align*}6x^2+18x-24\end{align*}
  11. \begin{align*}-n^2+10n-21\end{align*}
  12. \begin{align*}2a^2-14a-16\end{align*}

Review (Answers)

To view the Review answers, open this PDF file and look for section 9.12. 

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Vocabulary

TermDefinition
Cubed The cube of a number is the number multiplied by itself three times. For example, "two-cubed" = 2^3 = 2 \times 2 \times 2 = 8.
factor Factors are the numbers being multiplied to equal a product. To factor means to rewrite a mathematical expression as a product of factors.
Factor to Solve "Factor to Solve" is a common method for solving quadratic equations accomplished by factoring a trinomial into two binomials and identifying the values of x that make each binomial equal to zero.
Quadratic Formula The quadratic formula states that for any quadratic equation in the form ax^2+bx+c=0, x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}.
Trinomial A trinomial is a mathematical expression with three terms.

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