To divide the polynomial [tex]\( x^3 - 2x^2 + 16 \)[/tex] by the divisor [tex]\( -x + 2 \)[/tex], we need to perform polynomial long division. Here is the step-by-step solution:
1. Set up the division: Write [tex]\( x^3 - 2x^2 + 16 \)[/tex] under the division symbol and [tex]\(-x + 2\)[/tex] outside.
2. Divide the leading terms: Divide the leading term of the dividend [tex]\( x^3 \)[/tex] by the leading term of the divisor [tex]\(-x\)[/tex]. This gives [tex]\( \frac{x^3}{-x} = -x^2 \)[/tex].
3. Multiply and subtract: Multiply the entire divisor [tex]\(-x + 2\)[/tex] by [tex]\(-x^2\)[/tex] and subtract the result from the original polynomial.
[tex]\[
-x^2 \cdot (-x + 2) = x^3 - 2x^2
\][/tex]
Subtract [tex]\( x^3 - 2x^2 \)[/tex] from [tex]\( x^3 - 2x^2 + 16 \)[/tex]:
[tex]\[
(x^3 - 2x^2 + 16) - (x^3 - 2x^2) = 16
\][/tex]
4. Result after subtraction: The result after subtraction is simply [tex]\( 16 \)[/tex].
After completing the division process, we find that the quotient is [tex]\( -x^2 \)[/tex] and the remainder is [tex]\( 16 \)[/tex].
Thus, the final quotient and remainder from the division of [tex]\( x^3 - 2x^2 + 16 \)[/tex] by [tex]\(-x + 2\)[/tex] are:
[tex]\[ \text{Quotient: } -x^2 \][/tex]
[tex]\[ \text{Remainder: } 16 \][/tex]
