Sure! Let's perform the polynomial long division step-by-step.
We need to divide the polynomial [tex]\( x^3 - 5x^2 + 2x + 8 \)[/tex] by [tex]\( 2x + 8 \)[/tex].
1. Step 1: Dividing the Leading Term
- Identify the leading term of the dividend: [tex]\( x^3 \)[/tex].
- Identify the leading term of the divisor: [tex]\( 2x \)[/tex].
- Divide these leading terms: [tex]\( \frac{x^3}{2x} = \frac{x^2}{2} \)[/tex].
2. Step 2: Multiply and Subtract
- Multiply [tex]\( \frac{x^2}{2} \)[/tex] by the entire divisor [tex]\( 2x + 8 \)[/tex]:
[tex]\[
\frac{x^2}{2}(2x + 8) = x^3 + 4x^2
\][/tex]
- Subtract this result from the original polynomial:
[tex]\[
(x^3 - 5x^2 + 2x + 8) - (x^3 + 4x^2) = -9x^2 + 2x + 8
\][/tex]
3. Step 3: Bring Down the Next Term and Repeat
- Divide the new leading term: [tex]\( \frac{-9x^2}{2x} = \frac{-9x}{2} \)[/tex].
- Multiply [tex]\( \frac{-9x}{2} \)[/tex] by [tex]\( 2x + 8 \)[/tex]:
[tex]\[
\frac{-9x}{2}(2x + 8) = -9x^2 - 36x
\][/tex]
- Subtract:
[tex]\[
(-9x^2 + 2x + 8) - (-9x^2 - 36x) = 38x + 8
\][/tex]
4. Step 4: Repeat Again
- Divide the new leading term: [tex]\( \frac{38x}{2x} = 19 \)[/tex].
- Multiply [tex]\( 19 \)[/tex] by [tex]\( 2x + 8 \)[/tex]:
[tex]\[
19(2x + 8) = 38x + 152
\][/tex]
- Subtract:
[tex]\[
(38x + 8) - (38x + 152) = -144
\][/tex]
After these steps, we have determined that the quotient is:
[tex]\[ \frac{x^2}{2} - \frac{9x}{2} + 19 \][/tex]
And the remainder is:
[tex]\[ -144 \][/tex]
Therefore, the result of the division of the polynomial [tex]\( x^3 - 5x^2 + 2x + 8 \)[/tex] by [tex]\( 2x + 8 \)[/tex] is:
[tex]\[ \text{Quotient: } \frac{x^2}{2} - \frac{9x}{2} + 19, \quad \text{Remainder: } -144 \][/tex]
