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].
