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Answer :
To find the quotient of the given polynomials [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] and [tex]\((x^3 - 3)\)[/tex], we perform polynomial division, similar to long division but with polynomials. Here’s how you can understand the process:
1. Set up the division: The polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] is the dividend, and [tex]\(x^3 - 3\)[/tex] is the divisor.
2. Divide the leading term: Look at the leading term of the dividend, [tex]\(x^4\)[/tex], and the leading term of the divisor, [tex]\(x^3\)[/tex]. Divide these to find the first term of the quotient:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
3. Multiply and subtract: Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
x(x^3 - 3) = x^4 - 3x
\][/tex]
Subtract this from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 15
\][/tex]
4. Repeat the process: With the new polynomial [tex]\(5x^3 - 15\)[/tex], divide the leading term by the leading term of the divisor:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
Multiply the entire divisor by this term:
[tex]\[
5(x^3 - 3) = 5x^3 - 15
\][/tex]
Subtract:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
5. No remainder: This subtraction leaves no remainder, confirming that the division process is complete.
The quotient of the division is therefore:
[tex]\[
x + 5
\][/tex]
This is the quotient when dividing [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex], matching one of the given options.
1. Set up the division: The polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] is the dividend, and [tex]\(x^3 - 3\)[/tex] is the divisor.
2. Divide the leading term: Look at the leading term of the dividend, [tex]\(x^4\)[/tex], and the leading term of the divisor, [tex]\(x^3\)[/tex]. Divide these to find the first term of the quotient:
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
3. Multiply and subtract: Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
x(x^3 - 3) = x^4 - 3x
\][/tex]
Subtract this from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 15
\][/tex]
4. Repeat the process: With the new polynomial [tex]\(5x^3 - 15\)[/tex], divide the leading term by the leading term of the divisor:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
Multiply the entire divisor by this term:
[tex]\[
5(x^3 - 3) = 5x^3 - 15
\][/tex]
Subtract:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
5. No remainder: This subtraction leaves no remainder, confirming that the division process is complete.
The quotient of the division is therefore:
[tex]\[
x + 5
\][/tex]
This is the quotient when dividing [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex], matching one of the given options.
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