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Find the quotient and remainder using synthetic division.

[tex]\[
\begin{array}{l}
\quad P(x) = x^3 + 9x^2 - 8, \quad D(x) = x + 9 \\
\text{quotient:} \quad \square x \\
\text{remainder:} \quad \square
\end{array}
\][/tex]

Use the Remainder Theorem to evaluate [tex]\( P(-9) \)[/tex].

[tex]\[
P(-9) = \square
\][/tex]

Sagot :

GinnyAnswer
To find the quotient and remainder using synthetic division for the polynomial [tex]\( P(x) = x^3 + 9x^2 - 8 \)[/tex] with the divisor [tex]\( D(x) = x + 9 \)[/tex], and to evaluate [tex]\( P(-9) \)[/tex] using the remainder theorem, follow these steps:

### Synthetic Division

1. Identify the root of the divisor. Since [tex]\( D(x) = x + 9 \)[/tex], the root of the divisor is [tex]\( -9 \)[/tex].

2. Write down the coefficients of the polynomial [tex]\( P(x) \)[/tex] in descending order of power. For [tex]\( x^3 + 9x^2 - 8 \)[/tex], the coefficients are [tex]\( [1, 9, 0, -8] \)[/tex].

3. Perform synthetic division.
- Write down the root of the divisor, which is [tex]\( -9 \)[/tex].
- Begin by bringing down the leading coefficient, which is [tex]\( 1 \)[/tex].

The process:
- Multiply the root [tex]\( -9 \)[/tex] by the leading coefficient [tex]\( 1 \)[/tex] and add it to the next coefficient.
- Continue this process for each coefficient in the polynomial.

Steps:
- Start with [tex]\( 1 \)[/tex] (the leading coefficient).
- [tex]\( 1 \times (-9) = -9 \)[/tex]. Add this to the next coefficient [tex]\( 9 \)[/tex]: [tex]\( 9 + (-9) = 0 \)[/tex].
- [tex]\( 0 \times (-9) = 0 \)[/tex]. Add this to the next coefficient [tex]\( 0 \)[/tex]: [tex]\( 0 + 0 = 0 \)[/tex].
- [tex]\( 0 \times (-9) = 0 \)[/tex]. Add this to the next coefficient [tex]\( -8 \)[/tex]: [tex]\( -8 + 0 = -8 \)[/tex].

The results of these operations give us the coefficients of the quotient and the remainder.

4. The quotient and remainder.
- The quotient coefficients are obtained from the results of the synthetic division excluding the last value.
- The last value is the remainder.

Thus, we have:
- Quotient coefficients: [tex]\( [1, 0, 0] \)[/tex], which represents [tex]\( x^2 + 0x + 0 \)[/tex] or simply [tex]\( x^2 \)[/tex].
- Remainder: [tex]\( -8 \)[/tex].

### Use the Remainder Theorem to Evaluate [tex]\( P(-9) \)[/tex]

According to the Remainder Theorem, the value of [tex]\( P(-9) \)[/tex] is equal to the remainder obtained from the synthetic division.

Thus, [tex]\( P(-9) = -8 \)[/tex].

### Summary
- The quotient is [tex]\( x^2 \)[/tex].
- The remainder is [tex]\( -8 \)[/tex].
- [tex]\( P(-9) = -8 \)[/tex].
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