Answer :
To find the quotient of the division of the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by the polynomial [tex]\(x^3 - 3\)[/tex], we'll use polynomial long division. Here's a step-by-step explanation:
1. Set up the division:
Divide the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] (the numerator) by [tex]\(x^3 - 3\)[/tex] (the denominator).
2. Divide the leading term:
Look at the leading terms of both the dividend [tex]\(x^4\)[/tex] and the divisor [tex]\(x^3\)[/tex].
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
Write [tex]\(x\)[/tex] above the division bar.
3. Multiply and subtract:
Multiply [tex]\(x\)[/tex] by [tex]\(x^3 - 3\)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
Subtract this from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + (0)x - 15
\][/tex]
4. Bring down the next terms:
Since we subtracted correctly, we now focus on [tex]\(5x^3\)[/tex] (which remains) and the constant [tex]\(-15\)[/tex].
5. Divide the next leading term:
Divide the new leading term [tex]\(5x^3\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
Write [tex]\(5\)[/tex] next to the [tex]\(x\)[/tex] above the division bar, making it [tex]\(x + 5\)[/tex].
6. Multiply and subtract again:
Multiply [tex]\(5\)[/tex] by [tex]\(x^3 - 3\)[/tex]:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
Subtract this from what remains of the polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
7. Conclusion:
Since the remainder is 0, the division yields a quotient of [tex]\(x + 5\)[/tex].
Thus, the quotient of the division of [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
1. Set up the division:
Divide the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] (the numerator) by [tex]\(x^3 - 3\)[/tex] (the denominator).
2. Divide the leading term:
Look at the leading terms of both the dividend [tex]\(x^4\)[/tex] and the divisor [tex]\(x^3\)[/tex].
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
Write [tex]\(x\)[/tex] above the division bar.
3. Multiply and subtract:
Multiply [tex]\(x\)[/tex] by [tex]\(x^3 - 3\)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
Subtract this from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + (0)x - 15
\][/tex]
4. Bring down the next terms:
Since we subtracted correctly, we now focus on [tex]\(5x^3\)[/tex] (which remains) and the constant [tex]\(-15\)[/tex].
5. Divide the next leading term:
Divide the new leading term [tex]\(5x^3\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
Write [tex]\(5\)[/tex] next to the [tex]\(x\)[/tex] above the division bar, making it [tex]\(x + 5\)[/tex].
6. Multiply and subtract again:
Multiply [tex]\(5\)[/tex] by [tex]\(x^3 - 3\)[/tex]:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
Subtract this from what remains of the polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
7. Conclusion:
Since the remainder is 0, the division yields a quotient of [tex]\(x + 5\)[/tex].
Thus, the quotient of the division of [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
