Can you factor the following polynomial completely?

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

### Watch This

Khan Academy Factoring and the Distributive Property

### 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:

- Check to see if the cubic has any common factors. If it does, factor them out.
- 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
is a polynomial of degree equal to 3. For example \begin{align*}8x^3+2x^2-5x-7\end{align*} is a cubic polynomial.*cubic polynomial*

- Distributive Property
- The
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*}.*distributive property*

- Factor
- To
means to rewrite an expression as a product of other expressions. These resulting expressions are called the*factor*of the original expression.*factors*

- Factor Completely
- To
means to factor an expression until none of its factors can be factored any further.*factor completely*

- Greatest Common Factor
- The
(or*Greatest Common Factor*) is the largest monomial that is a factor of (or divides into evenly) each of the terms of the polynomial.*GCF*

### 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*}.

**Answers:**

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.

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

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 7.9.