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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. \begin{align*}6x^2-30x+36\end{align*}

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

\begin{align*}6(x^2-5x+6)\end{align*}

There are no special products. We factor \begin{align*}x^2-5x+6\end{align*} as a product of two binomials: \begin{align*}(x \ )(x \ )\end{align*}

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

\begin{align*}6(x^2-5x+6)=6(x-2)(x-3)\end{align*}

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

The answer is \begin{align*}6(x-2)(x-3)\end{align*}.

 

2. \begin{align*}2x^2-8\end{align*}

Factor out common monomials: \begin{align*}2x^2-8=2(x^2-4)\end{align*}

We recognize \begin{align*}x^2-4\end{align*} as a difference of squares. We factor it as \begin{align*}(x+2)(x-2)\end{align*}.

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

The answer is \begin{align*}2(x+2)(x-2)\end{align*}.

 

3. \begin{align*}x^3+6x^2+9x\end{align*}

Factor out common monomials: \begin{align*}x^3+6x^2+9x=x(x^2+6x+9)\end{align*}

We recognize \begin{align*}x^2+6x+9\end{align*} as a perfect square and factor it as \begin{align*}(x+3)^2\end{align*}.

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

The answer is \begin{align*}x(x+3)^2\end{align*}.

 

4. \begin{align*}-2x^4+162\end{align*}

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.)

\begin{align*}-2x^4+162=-2(x^4-81)\end{align*}

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

\begin{align*}-2(x^2-9)(x^2+9)\end{align*}

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

\begin{align*}-2(x+3)(x-3)(x^2+9)\end{align*}

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

The answer is \begin{align*}-2(x+3)(x-3)(x^2+9)\end{align*}.

 

5. \begin{align*}x^5-8x^3+16x\end{align*}

Factor out the common monomial: \begin{align*}x^5-8x^3+14x=x(x^4-8x^2+16)\end{align*}

We recognize \begin{align*}x^4-8x^2+16\end{align*} as a perfect square and we factor it as \begin{align*}x(x^2-4)^2\end{align*}.

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

We factor it and get \begin{align*}((x+2)(x-2))^2\end{align*}, which we can rewrite as \begin{align*}(x+2)^2(x-2)^2\end{align*}.

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

The final answer is \begin{align*}x(x+2)^2(x-2)^2\end{align*}.

 

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:

\begin{align*}x(3x+2)-5(3x+2)\end{align*}

Since the term \begin{align*}(3x+2)\end{align*} 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:

\begin{align*}(3x+2)(x-5)\end{align*}

This expression is now completely factored.

Factoring out Common Binomials - Learn by Example

1. \begin{align*}3x(x-1)+4(x-1)\end{align*}

\begin{align*}3x(x-1)+4(x-1)\end{align*} has a common binomial of \begin{align*}(x-1)\end{align*}.

When we factor out the common binomial we get \begin{align*}(x-1)(3x+4)\end{align*}.

 

2. \begin{align*}x(4x+5)+(4x+5)\end{align*}

\begin{align*}x(4x+5)+(4x+5)\end{align*} has a common binomial of \begin{align*}(4x+5)\end{align*}.

When we factor out the common binomial we get \begin{align*}(4x+5)(x+1)\end{align*}.

 

Example

Factor completely: \begin{align*}24x^3-28x^2+8x\end{align*}.

First, notice that each term has \begin{align*}4x\end{align*} as a factor. Start by factoring out \begin{align*}4x\end{align*}:

\begin{align*}24x^3-28x^2+8x=4x(6x^2-7x+2)\end{align*}

Next, factor the trinomial in the parenthesis. Since \begin{align*}a\neq 1\end{align*} find \begin{align*}a\cdot c\end{align*}: \begin{align*} 6\cdot 2=12\end{align*}. Find the factors of 12 that add up to -7. Since 12 is positive and -7 is negative, the two factors should be negative:

\begin{align*} 12&=-1\cdot -12 && \text{ and } && -1+-12=-13\\ 12&=-2\cdot -6 && \text{ and } && -2+-6=-8\\ 12&=-3\cdot -4 && \text{ and } && -3+-4=-7\end{align*}

Rewrite the trinomial using \begin{align*}-7x=-3x-4x\end{align*}, and then factor by grouping:

\begin{align*}6x^2-7x+2 &= 6x^2-3x-4x+2\\ &=3x(2x-1)-2(2x-1)\\ &=(3x-2)(2x-1)\end{align*}

The final factored answer is:

\begin{align*}4x(3x-2)(2x-1)\end{align*}

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

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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