Sure, let's divide [tex]\( x^3 + 2x^2 - 30x - 75 \)[/tex] by [tex]\( x + 5 \)[/tex].
### Step-by-Step Solution
1. Set Up the Division:
Divide [tex]\( x^3 + 2x^2 - 30x - 75 \)[/tex] by [tex]\( x + 5 \)[/tex].
2. Determine the First Term of the Quotient:
- To determine the first term of the quotient, divide the leading term of the dividend [tex]\( x^3 \)[/tex] by the leading term of the divisor [tex]\( x \)[/tex]:
[tex]\[
\frac{x^3}{x} = x^2
\][/tex]
- Write [tex]\( x^2 \)[/tex] above the long division bar.
3. Multiply and Subtract:
- Multiply [tex]\( x^2 \)[/tex] by [tex]\( x + 5 \)[/tex]:
[tex]\[
x^2 \cdot (x + 5) = x^3 + 5x^2
\][/tex]
- Subtract this result from the original polynomial:
[tex]\[
(x^3 + 2x^2 - 30x - 75) - (x^3 + 5x^2) = (2x^2 - 5x^2) - 30x - 75 = -3x^2 - 30x - 75
\][/tex]
4. Determine the Next Term of the Quotient:
- Divide the new leading term [tex]\(-3x^2\)[/tex] by the leading term of the divisor [tex]\( x \)[/tex]:
[tex]\[
\frac{-3x^2}{x} = -3x
\][/tex]
- Write [tex]\(-3x\)[/tex] above the long division bar next to [tex]\( x^2 \)[/tex].
5. Multiply and Subtract:
- Multiply [tex]\(-3x\)[/tex] by [tex]\( x + 5 \)[/tex]:
[tex]\[
-3x \cdot (x + 5) = -3x^2 - 15x
\][/tex]
- Subtract this result from the current polynomial:
[tex]\[
(-3x^2 - 30x - 75) - (-3x^2 - 15x) = (-30x + 15x) - 75 = -15x - 75
\][/tex]
6. Determine the Last Term of the Quotient:
- Divide the new leading term [tex]\(-15x\)[/tex] by the leading term of the divisor [tex]\( x \)[/tex]:
[tex]\[
\frac{-15x}{x} = -15
\][/tex]
- Write [tex]\(-15\)[/tex] above the long division bar next to [tex]\(-3x\)[/tex].
7. Multiply and Subtract:
- Multiply [tex]\(-15\)[/tex] by [tex]\( x + 5 \)[/tex]:
[tex]\[
-15 \cdot (x + 5) = -15x - 75
\][/tex]
- Subtract this result from the current polynomial:
[tex]\[
(-15x - 75) - (-15x - 75) = 0
\][/tex]
8. Conclusion:
- The quotient is [tex]\( x^2 - 3x - 15 \)[/tex].
- The remainder is [tex]\( 0 \)[/tex].
Hence, when we divide [tex]\( x^3 + 2x^2 - 30x - 75 \)[/tex] by [tex]\( x + 5 \)[/tex], we get:
[tex]\[
\text{Quotient} = x^2 - 3x - 15, \quad \text{Remainder} = 0
\][/tex]
