The volume of a rectangular prism is

### Watch This

James Sousa: Polynomial Division: Synthetic Division

### Guidance

Synthetic division is an alternative to long division from the previous concept. It can also be used to divide a polynomial by a possible factor,

#### Example A

Divide

**Solution:** Using synthetic division, the setup is as follows:

To “read” the answer, use the numbers as follows:

Therefore, 2 is a solution, because the remainder is zero. The factored polynomial is

#### Example B

Determine if 4 is a solution to

Using synthetic division, we have:

The remainder is 304, so 4 is not a solution. Notice if we substitute in

**Remainder Theorem:** If

This means that if you substitute in

#### Example C

Determine if

**Solution:** If you use synthetic division, the factor is not in the form

This means that

#### Intro Problem Revisit

If

If we want to use synthetic division, notice that the factor is not in the form

When we perform the synthetic division, we get a remainder of 0. This means that

### Guided Practice

1. Divide

2. Divide

3. Is 6 a solution for

#### Answers

1. Using synthetic division, divide by -3.

The answer is

2. Using synthetic division, divide by

The answer is

3. Put a zero placeholder for the

The resulting polynomial is

The solutions to this polynomial are 6,

### Explore More

Use synthetic division to divide the following polynomials. Write out the remaining polynomial.

- \begin{align*}(x^3+6x^2+7x+10) \div (x+2)\end{align*}
- \begin{align*}(4x^3-15x^2-120x-128) \div (x-8)\end{align*}
- \begin{align*}(4x^2-5) \div (2x+1)\end{align*}
- \begin{align*}(2x^4-15x^3-30x^2-20x+42) \div (x+9)\end{align*}
- \begin{align*}(x^3-3x^2-11x+5) \div (x-5)\end{align*}
- \begin{align*}(3x^5+4x^3-x-2) \div (x-1)\end{align*}
- Which of the division problems above generate no remainder? What does that mean?
- What is the difference between a zero and a factor?
- Find \begin{align*}f(-2)\end{align*} if \begin{align*}f(x)=2x^4-5x^3-10x^2+21x-4\end{align*}.
- Now, divide \begin{align*}2x^4-5x^3-10x^2+21x-4\end{align*} by \begin{align*}(x + 2)\end{align*} synthetically. What do you notice?

Find all real zeros of the following polynomials, given one zero.

- \begin{align*}12x^3+76x^2+107x-20; -4\end{align*}
- \begin{align*}x^3-5x^2-2x+10; -2\end{align*}
- \begin{align*}6x^3-17x^2+11x-2; 2\end{align*}

Find all real zeros of the following polynomials, given two zeros.

- \begin{align*}x^4+7x^3+6x^2-32x-32; -4, -1\end{align*}
- \begin{align*}6x^4+19x^3+11x^2-6x; 0, -2\end{align*}