Question

Use synthetic division and the remainder theorem to find P(a).
P(x)=x^3 +5x^2 -7x+1; a=2
P(a)=____
(Simplify your answer)

Answer 1

To find \( P(2) \) using synthetic division, we will use the coefficients of the polynomial \( P(x) = x^3 + 5x^2 - 7x + 1 \).

The coefficients are:
- \( 1 \) (for \( x^3 \))
- \( 5 \) (for \( x^2 \))
- \( -7 \) (for \( x \))
- \( 1 \) (constant term)

Now, we will perform synthetic division with \( a = 2 \).

1. Write down the coefficients:
\[
1 \quad 5 \quad -7 \quad 1
\]

2. Draw a line and write \( 2 \) to the left:
\[
2 \quad | \quad 1 \quad 5 \quad -7 \quad 1
\]

3. Bring down the \( 1 \) (the leading coefficient):
\[
2 \quad | \quad 1 \quad 5 \quad -7 \quad 1 \\
\quad | \quad \downarrow \\
\quad | \quad 1
\]

4. Multiply \( 2 \) (the number to the left) by \( 1 \) (the number you just brought down):
\( 2 \times 1 = 2 \)

5. Add this result to the next coefficient (\( 5 \)):
\[
5 + 2 = 7
\]
Now the synthetic division looks like this:
\[
2 \quad | \quad 1 \quad 5 \quad -7 \quad 1 \\
\quad | \quad \downarrow \quad 2 \\
\quad | \quad 1 \quad 7
\]

6. Repeat the process: multiply \( 2 \) by \( 7 \):
\( 2 \times 7 = 14 \)

7. Add this to the next coefficient (\( -7 \)):
\[
-7 + 14 = 7
\]
Now the synthetic division looks like this:
\[
2 \quad | \quad 1 \quad 5 \quad -7 \quad 1 \\
\quad | \quad \downarrow \quad 2 \quad 14 \\
\quad | \quad 1 \quad 7 \quad 7
\]

8. Finally, multiply \( 2 \) by \( 7 \):
\( 2 \times 7 = 14 \)

9. Add this to the last coefficient (\( 1 \)):
\[
1 + 14 = 15
\]

The last number obtained through synthetic division, \( 15 \), is \( P(2) \).

Thus, we have:
\[
P(2) = 15.
\]

Therefore,
\[
P(a) = 15.
\]