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AP Calculus AB & BC: Help and Review17 chapters | 160 lessons

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Lesson Transcript

Instructor:
*Shaun Ault*

Shaun is currently an Assistant Professor of Mathematics at Valdosta State University as well as an independent private tutor.

This lesson is about synthetic division. Synthetic division helps us work out polynomial division when the divisor is simple. It also is a great shortcut to finding roots of polynomials.

We all love shortcuts! **Synthetic division** is a shortcut method for dividing a polynomial by a simple divisor of the form (*x* - *n*). The divisor must be of that form in order for synthetic division to work. If it's not, you'll have to use long division.

So why would we want to divide polynomials anyway? If a polynomial *p*(*x*) can be divided by (*x* - *n*) with no remainder, then *x* = *n* must be a zero or root of *p*(*x*). That is, a solution to the equation *p*(*x*) = 0. We'll see how that works in some examples below.

Synthetic division uses only the **coefficients** of a polynomial, or the constants in front of each *x*-term, so it saves a ton of writing compared to using long division of polynomials.

There is a specific order of steps to synthetic division; once you get the pattern, you will be off and running! The best way to explain these steps is by way of an example. Let's use synthetic division to work out this problem:

(3*x*^3 - *x* - 7) / (*x* - 2)

Step 1. If the divisor is (*x* - *n*), write *n*, and then draw a vertical line to the right of it. In this example, *n* = 2, so we would write:

2 |

Step 2. Put the coefficients of each *x*-term to the right of the vertical line. Start with the **leading coefficient**, that is, the coefficient of the highest power term, then place the coefficients of each lower degree term in descending order. It's very important to place a number for each degree, so if a particular term *x*^d does not show up, you should place a 0 in that spot. In our example, we could interpret 3*x*^3 - *x* - 7 = 3*x*^3 + 0*x*^2 + (-1)*x* + (-7). So the first line of your synthetic division work should look like this:

2 | 3 0 -1 -7

Step 3. Draw a horizontal line under the coefficients, leaving one empty row below the coefficients for work.

Step 4. Bring the first coefficient down below the horizontal line, which is the easiest step of all!

Step 5. Multiply the divisor number *n* by the number below the horizontal line, and place the result above the line below the next coefficient. In our running example. 2 * 3 = 6, which goes right below the 0.

Step 6. Add the column to get the next coefficient in your answer. In our example, that would be 0 + 6 = 6. This new number goes below the horizontal line.

Step 7. Repeat steps 5 and 6, filling each column from left to right until you get to the end of the coefficients. Do you know how the rest of the steps work out?

Circle or otherwise mark in some way the very last sum in the far right column. That number is the **remainder** , or the number left over after dividing.

The numbers below the horizontal line that have not been circled are the coefficients of your answer, the **quotient**. Each coefficient belongs to an *x* term having exactly 1 less degree than the corresponding coefficient above it. If there was a nonzero remainder, say *r*, then tack on + *r* / (*x* - *n*) to the end of your quotient.

In our example, the leading term has degree 3, so the quotient must begin with one less degree, an *x*^2 term. The final answer, including the quotient and remainder, should be written as follows:

(3*x*^3 - *x* - 7) / (*x* - 2) = 3*x*^2 + 6*x* + 11 + 15 / (*x* - 2).

Practice makes perfect! Let's work out another example together.

Divide (3*x*^4 - 2*x*^2 + 4) / (*x* + 1).

Note that the first polynomial, the **dividend**, is missing two terms, *x*^3 and *x*. Thus, the coefficients in descending order of degree are: 3, 0, -2, 0, 4. Also, *n* = -1 because (*x* + 1) = (*x* - (-1)) . The first row looks like this:

-1 | 3 0 -2 0 4

Here's an image of the finished table and an outline and the steps are outlined below.

- Bring the 3 down
- Multiply (-1) * 3 = -3.
- Add 0 + (-3) = -3.
- Multiply (-1) * (-3) = 3.
- Add (-2) + 3 = 1.
- Multiply (-1) * 1 = -1.
- Add 0 + (-1) = -1.
- Multiply (-1) * (-1) = 1.
- Add 4 + 1 = 5.

Since the dividend has degree 4, the quotient must have degree 3. Remember, the last column shows your remainder. The final answer is:

3*x*^3 - 3*x*^2 + *x* - *1* + 5/(*x* + 1).

If there's no remainder after polynomial division, *p*(*x*) / (*x* - *n*), then you know that (*x* - *n*) is a **factor** of *p*(*x*), and for that reason, *x* = *n* is a root of the polynomial.

For example, let *p*(*x*) = *x*^3 - 3*x*^2 - 4*x* + 12. It turns out that *x* = 1 is not a root, because there is a nonzero remainder in the synthetic division:

However, *x* = 2 is definitely a root!

**Synthetic Division** is a shortcut method for dividing polynomials and finding the zeros of the polynomial. The method works only if the divisor has the form (*x* - *n*). If the remainder turns out to be 0, then *n* is a **root** of the polynomial.

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AP Calculus AB & BC: Help and Review17 chapters | 160 lessons

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