# 6.10: Synthetic Division of Polynomials

**At Grade**Created by: CK-12

**Practice**Synthetic Division of Polynomials

The volume of a rectangular prism is \begin{align*}2x^3 + 5x^2 - x - 6\end{align*}

### 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, \begin{align*}x - k\end{align*}

#### Example A

Divide \begin{align*}2x^4-5x^3-14x^2-37x-30\end{align*}

**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 \begin{align*}2x^3-x^2-16x+15\end{align*}

#### Example B

Determine if 4 is a solution to \begin{align*}f(x)=5x^3+6x^2-24x-16\end{align*}

Using synthetic division, we have:

The remainder is 304, so 4 is not a solution. Notice if we substitute in \begin{align*}x = 4\end{align*}

**Remainder Theorem:** If \begin{align*}f(k) = r\end{align*}

This means that if you substitute in \begin{align*}x = k\end{align*}

#### Example C

Determine if \begin{align*}(2x - 5)\end{align*}

**Solution:** If you use synthetic division, the factor is not in the form \begin{align*}(x - k)\end{align*}

This means that \begin{align*}\frac{5}{2}\end{align*}

**Intro Problem Revisit**

If \begin{align*}2x + 3\end{align*}

If we use synthetic division, the factor is not in the form \begin{align*}(x - k)\end{align*}

When we perform the synthetic division, we get a remainder of 0. This means that \begin{align*}(2x + 3)\end{align*}

### Guided Practice

1. Divide \begin{align*}x^3-9x^2+12x-27\end{align*}

2. Divide \begin{align*}2x^4-11x^3+12x^2+9x-2\end{align*}

3. Is 6 a solution for \begin{align*}f(x)=x^3-8x^2+72\end{align*}

#### Answers

1. Using synthetic division, divide by -3.

The answer is \begin{align*}x^2+6x-6-\frac{9}{x+3}\end{align*}

2. Using synthetic division, divide by \begin{align*}-\frac{1}{2}\end{align*}

The answer is \begin{align*}2x^3-12x^2+18x-\frac{2}{2x+1}\end{align*}

3. Put a zero placeholder for the \begin{align*}x-\end{align*}

The resulting polynomial is \begin{align*}x^2-2x-12\end{align*}

\begin{align*}x=\frac{2 \pm \sqrt{2^2-4(1)(-12)}}{2}=\frac{2 \pm \sqrt{4+48}}{2}=\frac{2 \pm 2 \sqrt{13}}{2}=1 \pm \sqrt{13}\end{align*}

The solutions to this polynomial are 6, \begin{align*}1+\sqrt{13} \approx 4.61\end{align*} and \begin{align*}1-\sqrt{13} \approx -2.61\end{align*}.

### Vocabulary

- Synthetic Division
- An alternative to long division for dividing \begin{align*}f(x)\end{align*} by \begin{align*}k\end{align*} where only the coefficients of \begin{align*}f(x)\end{align*} are used.

- Remainder Theorem
- If \begin{align*}f(k) = r\end{align*}, then \begin{align*}r\end{align*} is also the remainder when dividing by \begin{align*}(x - k)\end{align*}.

### Practice

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-6; 0, -2\end{align*}

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Oblique Asymptote

An oblique asymptote is a diagonal line marking a specific range of values toward which the graph of a function may approach, but generally never reach. An oblique asymptote exists when the numerator of the function is exactly one degree greater than the denominator. An oblique asymptote may be found through long division.Oblique Asymptotes

An oblique asymptote is a diagonal line marking a specific range of values toward which the graph of a function may approach, but generally never reach. An oblique asymptote exists when the numerator of the function is exactly one degree greater than the denominator. An oblique asymptote may be found through long division.Remainder Theorem

The remainder theorem states that if , then is the remainder when dividing by .Synthetic Division

Synthetic division is a shorthand version of polynomial long division where only the coefficients of the polynomial are used.### Image Attributions

Here you'll learn how to use synthetic division as a short-cut and alternative to long division (in certain cases) and to find zeros.