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Answer :
To find the quotient of the polynomial division of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex], we can perform polynomial long division.
Here's a step-by-step process:
1. Divide the first term of the dividend by the first term of the divisor:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
So, the first term of the quotient is [tex]\(x\)[/tex].
2. Multiply the entire divisor by this term ([tex]\(x\)[/tex]) and subtract from the original dividend:
Multiply:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
Subtract:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15
\][/tex]
3. Repeat the process with the new polynomial (i.e., [tex]\(5x^3 - 15\)[/tex]):
Divide the first term:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
So, the next term in the quotient is [tex]\(5\)[/tex].
4. Multiply the entire divisor by this term ([tex]\(5\)[/tex]) and subtract:
Multiply:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
Subtract:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Since we are left with a remainder of zero, the process stops here, and the quotient of the division is given by the terms we found:
The quotient is [tex]\(x + 5\)[/tex].
So, the answer is [tex]\(x + 5\)[/tex].
Here's a step-by-step process:
1. Divide the first term of the dividend by the first term of the divisor:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
So, the first term of the quotient is [tex]\(x\)[/tex].
2. Multiply the entire divisor by this term ([tex]\(x\)[/tex]) and subtract from the original dividend:
Multiply:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
Subtract:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15
\][/tex]
3. Repeat the process with the new polynomial (i.e., [tex]\(5x^3 - 15\)[/tex]):
Divide the first term:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
So, the next term in the quotient is [tex]\(5\)[/tex].
4. Multiply the entire divisor by this term ([tex]\(5\)[/tex]) and subtract:
Multiply:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
Subtract:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Since we are left with a remainder of zero, the process stops here, and the quotient of the division is given by the terms we found:
The quotient is [tex]\(x + 5\)[/tex].
So, the answer is [tex]\(x + 5\)[/tex].
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