To find the quotient [tex]\( Q(x) \)[/tex] and the remainder [tex]\( R(x) \)[/tex] when [tex]\( P(x) = x^3 + 5x^2 - 10x + 114 \)[/tex] is divided by [tex]\( d(x) = x + 8 \)[/tex], we use polynomial long division. Here's the detailed step-by-step solution:
1. Set up the division:
We divide [tex]\( x^3 + 5x^2 - 10x + 114 \)[/tex] by [tex]\( x + 8 \)[/tex].
2. Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{x^3}{x} = x^2
\][/tex]
So, the first term of the quotient is [tex]\( x^2 \)[/tex].
3. Multiply the entire divisor by this term [tex]\( x^2 \)[/tex]:
[tex]\[
(x + 8)x^2 = x^3 + 8x^2
\][/tex]
4. Subtract this product from the original polynomial [tex]\( P(x) \)[/tex]:
[tex]\[
(x^3 + 5x^2 - 10x + 114) - (x^3 + 8x^2) = -3x^2 - 10x + 114
\][/tex]
5. Repeat the process with the new polynomial [tex]\( -3x^2 - 10x + 114 \)[/tex]:
* Divide the leading term of the new polynomial by the leading term of the divisor:
[tex]\[
\frac{-3x^2}{x} = -3x
\][/tex]
So, the next term of the quotient is [tex]\(-3x\)[/tex].
* Multiply the divisor by [tex]\(-3x\)[/tex]:
[tex]\[
(x + 8)(-3x) = -3x^2 - 24x
\][/tex]
* Subtract this product from the new polynomial:
[tex]\[
(-3x^2 - 10x + 114) - (-3x^2 - 24x) = 14x + 114
\][/tex]
6. Repeat the process again with the new polynomial [tex]\( 14x + 114 \)[/tex]:
* Divide the leading term of the new polynomial by the leading term of the divisor:
[tex]\[
\frac{14x}{x} = 14
\][/tex]
So, the next term of the quotient is [tex]\( 14 \)[/tex].
* Multiply the divisor by [tex]\( 14 \)[/tex]:
[tex]\[
(x + 8)(14) = 14x + 112
\][/tex]
* Subtract this product from the new polynomial:
[tex]\[
(14x + 114) - (14x + 112) = 2
\][/tex]
7. Stop the division as the degree of the remainder [tex]\( 2 \)[/tex] is less than the degree of the divisor [tex]\( x + 8 \)[/tex].
Therefore, the quotient [tex]\( Q(x) \)[/tex] is:
[tex]\[
Q(x) = x^2 - 3x + 14
\][/tex]
And the remainder [tex]\( R(x) \)[/tex] is:
[tex]\[
R(x) = 2
\][/tex]
Finally, we express [tex]\( P(x) \)[/tex] in the form [tex]\( d(x) \cdot Q(x) + R(x) \)[/tex]:
[tex]\[
P(x) = (x + 8)(x^2 - 3x + 14) + 2
\][/tex]
So the completed form is:
[tex]\[
P(x) = (x + 8) \cdot (x^2 - 3x + 14) + 2
\][/tex]
