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Answer :
To find the quotient of the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], we perform polynomial division. Here's how it works step by step:
1. Set up the division: We have the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and the divisor [tex]\(x^3 - 3\)[/tex].
2. Divide the leading terms: Start by dividing the leading term of the dividend, [tex]\(x^4\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex]. This gives us the first term of the quotient: [tex]\(x\)[/tex].
3. Multiply and subtract: Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by this first term of the quotient, [tex]\(x\)[/tex]:
[tex]\[
x(x^3 - 3) = x^4 - 3x
\][/tex]
Subtract this result from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 15
\][/tex]
4. Repeat the process: Now, divide the new leading term [tex]\(5x^3\)[/tex] by the leading term of the divisor, [tex]\(x^3\)[/tex]. This gives us the next term of the quotient: [tex]\(+5\)[/tex].
5. Multiply and subtract again: Multiply the entire divisor by [tex]\(5\)[/tex]:
[tex]\[
5(x^3 - 3) = 5x^3 - 15
\][/tex]
Subtract this from the remaining polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Since the remainder is 0, the final quotient is [tex]\(x + 5\)[/tex].
Therefore, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is the polynomial [tex]\((x + 5)\)[/tex].
1. Set up the division: We have the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and the divisor [tex]\(x^3 - 3\)[/tex].
2. Divide the leading terms: Start by dividing the leading term of the dividend, [tex]\(x^4\)[/tex], by the leading term of the divisor, [tex]\(x^3\)[/tex]. This gives us the first term of the quotient: [tex]\(x\)[/tex].
3. Multiply and subtract: Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by this first term of the quotient, [tex]\(x\)[/tex]:
[tex]\[
x(x^3 - 3) = x^4 - 3x
\][/tex]
Subtract this result from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 15
\][/tex]
4. Repeat the process: Now, divide the new leading term [tex]\(5x^3\)[/tex] by the leading term of the divisor, [tex]\(x^3\)[/tex]. This gives us the next term of the quotient: [tex]\(+5\)[/tex].
5. Multiply and subtract again: Multiply the entire divisor by [tex]\(5\)[/tex]:
[tex]\[
5(x^3 - 3) = 5x^3 - 15
\][/tex]
Subtract this from the remaining polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Since the remainder is 0, the final quotient is [tex]\(x + 5\)[/tex].
Therefore, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is the polynomial [tex]\((x + 5)\)[/tex].
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