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Factoring Completely

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

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Complete Factorization of Polynomials

Can you factor the following polynomial completely?

\begin{align*}8x^3+24x^2-32x\end{align*}

Guidance

A cubic polynomial is a polynomial of degree equal to 3. Examples of cubics are:

• \begin{align*}9x^3+10x-5\end{align*}
• \begin{align*}8x^3+2x^2-5x-7\end{align*}

Recall that to factor a polynomial means to rewrite the polynomial as a product of other polynomials . You will not be able to factor all cubics at this point, but you will be able to factor some using your knowledge of common factors and factoring quadratics. In order to attempt to factor a cubic, you should:

1. Check to see if the cubic has any common factors. If it does, factor them out.
2. Check to see if the resulting expression can be factored, especially if the resulting expression is a quadratic. To factor the quadratic expression you could use the box method, or any method you prefer.

Anytime you are asked to factor completely, you should make sure that none of the pieces (factors) of your final answer can be factored any further. If you follow the steps above of first checking for common factors and then checking to see if the resulting expressions can be factored, you can be confident that you have factored completely.

Example A

Factor the following polynomial completely: \begin{align*}3x^3-15x\end{align*}.

Solution: Look for the common factors in each of the terms. The common factor is \begin{align*}3x\end{align*}. Therefore:

\begin{align*}3x^3-15x=3x(x^2-5)\end{align*}

The resulting quadratic, \begin{align*}x^2-5\end{align*}, cannot be factored any further (it is NOT a difference of perfect squares). Your answer is \begin{align*}3x(x^2-5)\end{align*}.

Example B

Factor the following polynomial completely: \begin{align*}2a^3+16a^2+30a\end{align*}.

Solution: Look for the common factors in each of the terms. The common factor is \begin{align*}2a\end{align*}. Therefore:

\begin{align*}2a^3+16a^2+30a=2a(a^2+8a+15)\end{align*}

The resulting quadratic, \begin{align*}a^2+8a+15\end{align*} can be factored further into \begin{align*}(a+5)(a+3)\end{align*}. Your final answer is \begin{align*}2a(a+5)(a+3)\end{align*}.

Example C

Factor the following polynomial completely: \begin{align*}6s^3+36s^2-18s-42\end{align*}.

Solution: Look for the common factors in each of the terms. The common factor is \begin{align*}6\end{align*}. Therefore:

\begin{align*}6s^3+36s^2-18s-42=6(s^3+6s^2-3s-7)\end{align*}

The resulting expression is a cubic, and you don't know techniques for factoring cubics without common factors at this point. Therefore, your final answer is \begin{align*}6(s^3+6s^2-3s-7).\end{align*}

Note: It turns out that the resulting cubic cannot be factored, even with more advanced techniques. Remember that not all expressions can be factored. In fact, in general most expressions cannot be factored.

Concept Problem Revisited

Factor the following polynomial completely: \begin{align*}8x^3+24x^2-32x\end{align*}.

Look for the common factors in each of the terms. The common factor is \begin{align*}8x\end{align*}. Therefore:

\begin{align*}8x^3 +24x^2 + 32x = 8x(x^2+3x-4)\end{align*}

The resulting quadratic can be factored further into \begin{align*}(x+4)(x-1)\end{align*}. Your final answer is \begin{align*}8x(x+4)(x-1)\end{align*}.

Vocabulary

Cubic Polynomial
A cubic polynomial is a polynomial of degree equal to 3. For example \begin{align*}8x^3+2x^2-5x-7\end{align*} is a cubic polynomial.
Distributive Property
The distributive property states that the product of a number and a sum is equal to the sum of the individual products of the number and the addends. For example, the distributive property says that \begin{align*}2(x+5)=2x+10\end{align*}.
Factor
To factor means to rewrite an expression as a product of other expressions. These resulting expressions are called the factors of the original expression.
Factor Completely
To factor completely means to factor an expression until none of its factors can be factored any further.
Greatest Common Factor
The Greatest Common Factor (or GCF) is the largest monomial that is a factor of (or divides into evenly) each of the terms of the polynomial.

Guided Practice

Factor each of the following polynomials completely.

1. \begin{align*}9w^3+12w\end{align*}.

2. \begin{align*}y^3+4y^2+4y\end{align*}.

3. \begin{align*}2t^3-10t^2+8t\end{align*}.

1. The common factor is \begin{align*}3w\end{align*}. Therefore, \begin{align*}9w^3+12w=3w(3w^2+4)\end{align*}. The resulting quadratic cannot be factored any further, so your answer is \begin{align*}3w(3w^2+4)\end{align*}.

2. The common factor is \begin{align*}y\end{align*}. Therefore, \begin{align*}y^3+4y^2+4y=y(y^2+4y+4)\end{align*}. The resulting quadratic can be factored into \begin{align*}(y+2)(y+2)\end{align*} or \begin{align*}(y+2)^2\end{align*}. Your answer is \begin{align*}y(y+2)^2\end{align*}.

3. The common factor is \begin{align*}2t\end{align*}. Therefore, \begin{align*}2t^3-10t^2+8t=2t(t^2-5t+4)\end{align*}. The resulting quadratic can be factored into \begin{align*}(t-4)(t-1)\end{align*}. Your answer is \begin{align*}2t(t-4)(t-1).\end{align*}

Practice

Factor each of the following polynomials completely.

1. \begin{align*}6x^3-12\end{align*}
2. \begin{align*}4x^3-12x^2\end{align*}
3. \begin{align*}8y^3+32y\end{align*}
4. \begin{align*}15a^3+30a^2\end{align*}
5. \begin{align*}21q^3+63q\end{align*}
6. \begin{align*}4x^3-12x^2-8\end{align*}
7. \begin{align*}12e^3+6e^2-6e\end{align*}
8. \begin{align*}15s^3-30s+45\end{align*}
9. \begin{align*}22r^3+66r^2+44r\end{align*}
10. \begin{align*}32d^3-16d^2+12d\end{align*}
11. \begin{align*}5x^3+15x^2+25x-30\end{align*}
12. \begin{align*}3y^3-18y^2+27y\end{align*}
13. \begin{align*}12s^3-24s^2+36s-48\end{align*}
14. \begin{align*}8x^3+24x^2-80x\end{align*}
15. \begin{align*}5x^3-25x^2-70x\end{align*}

Vocabulary Language: English

Cubed

Cubed

The cube of a number is the number multiplied by itself three times. For example, "two-cubed" = $2^3 = 2 \times 2 \times 2 = 8$.
distributive property

distributive property

The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, $a(b + c) = ab + ac$.
factor

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

"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.

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

Trinomial

A trinomial is a mathematical expression with three terms.