Answer :
To find the quotient of the division of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex], we can use polynomial long division. Let's go through the process step-by-step:
1. Divide the first term:
Divide the first term of the dividend [tex]\(x^4\)[/tex] by the first term of the divisor [tex]\(x^3\)[/tex]:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
2. Multiply and subtract:
Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by the result from step 1, which is [tex]\(x\)[/tex]:
[tex]\[
x(x^3 - 3) = x^4 - 3x
\][/tex]
Subtract this from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x - 15
\][/tex]
3. Repeat the process:
Divide the first term of the new dividend [tex]\(5x^3\)[/tex] by the first term of the divisor [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
4. Multiply and subtract again:
Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex]:
[tex]\[
5(x^3 - 3) = 5x^3 - 15
\][/tex]
Subtract this from the result of the previous subtraction:
[tex]\[
(5x^3 + 0x - 15) - (5x^3 - 15) = 0x + 0
\][/tex]
The result of the division is the polynomial [tex]\(x + 5\)[/tex] with no remainder, meaning the original polynomial divides evenly.
Therefore, the quotient is [tex]\(x + 5\)[/tex].
1. Divide the first term:
Divide the first term of the dividend [tex]\(x^4\)[/tex] by the first term of the divisor [tex]\(x^3\)[/tex]:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
2. Multiply and subtract:
Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by the result from step 1, which is [tex]\(x\)[/tex]:
[tex]\[
x(x^3 - 3) = x^4 - 3x
\][/tex]
Subtract this from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x - 15
\][/tex]
3. Repeat the process:
Divide the first term of the new dividend [tex]\(5x^3\)[/tex] by the first term of the divisor [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
4. Multiply and subtract again:
Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex]:
[tex]\[
5(x^3 - 3) = 5x^3 - 15
\][/tex]
Subtract this from the result of the previous subtraction:
[tex]\[
(5x^3 + 0x - 15) - (5x^3 - 15) = 0x + 0
\][/tex]
The result of the division is the polynomial [tex]\(x + 5\)[/tex] with no remainder, meaning the original polynomial divides evenly.
Therefore, the quotient is [tex]\(x + 5\)[/tex].
