Answer :
To find the quotient of the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], we can perform polynomial long division. Here's how it's done step-by-step:
1. Set up the division:
You have [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] as the dividend and [tex]\((x^3 - 3)\)[/tex] as the divisor.
2. Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
3. Multiply the entire divisor by this result and subtract:
- Multiply [tex]\((x^3 - 3)\)[/tex] by [tex]\(x\)[/tex]: [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 0x - 15
\][/tex]
4. Repeat the process with the new dividend (5x^3 - 15):
- Divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
- Multiply (x^3 - 3) by [tex]\(5\)[/tex]:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract from the current dividend:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
5. The remainder is 0, and you have found the quotient:
From steps 2 and 4, the terms you used ([tex]\(x\)[/tex] and [tex]\(5\)[/tex]), when combined, form the quotient:
[tex]\[
x + 5
\][/tex]
Thus, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex]. The answer is:
[tex]\[
x + 5
\][/tex]
1. Set up the division:
You have [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] as the dividend and [tex]\((x^3 - 3)\)[/tex] as the divisor.
2. Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
3. Multiply the entire divisor by this result and subtract:
- Multiply [tex]\((x^3 - 3)\)[/tex] by [tex]\(x\)[/tex]: [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract this from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 0x - 15
\][/tex]
4. Repeat the process with the new dividend (5x^3 - 15):
- Divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
- Multiply (x^3 - 3) by [tex]\(5\)[/tex]:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract from the current dividend:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
5. The remainder is 0, and you have found the quotient:
From steps 2 and 4, the terms you used ([tex]\(x\)[/tex] and [tex]\(5\)[/tex]), when combined, form the quotient:
[tex]\[
x + 5
\][/tex]
Thus, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex]. The answer is:
[tex]\[
x + 5
\][/tex]
