How do you use synthetic division to find P(4) for P(x)=x4+x3+10x2+9x−6?

1 Answer
Jim H
Jul 2, 2015

When P(x) is divided by x−4, the remainder is P(4).

Explanation:

(Synthetic Division Formatting by Truong-Son R.)

To find P(4) by division, divide x4+x3+10x2+9x−6 by x−4.
The Remainder Theorem to tells us that the remainder when we do the division will be equal to P(4)
Use synthetic division, because we've been told to. (And it is faster than long division.)

First, you let the coefficients of each degree be used in the division (1,1,10,1,−6).

Then, dividing by x−4 implies that you use 4 in your upper left. So, draw the bottom and right sides of a square, put 4 inside it, and then write 1 1 10 9 -6 to the right.

1∣∣ 1 1 10 9 -6
+
−−−−−

First, bring the first 1 down to the bottom, and multiply it by the 4. Put that 4 below the second 1.

4∣∣ 1 1 10 9 -6
+ 4
−−−−−
1

Then add it up:

4∣∣ 1 1 10 9 -6
+ 4
−−−−−
1 5

Multiply 4×5 and pout the 20 under the 10. Then add:

4∣∣ 1 1 10 9 -6
+ 4 20
−−−−−−−−
1 5 30

Repeat to get:

4∣∣ 1 1 10 9 -6
+ 4 20 120 516
−−−−−−−−−
1 5 30 129 510

The last number on the bottom row is the remainder and is also P(4), so P(4)=510

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