Answer :
Sure, let's solve the problem step by step to find the quotient of [tex]\(\left(x^4 + 5x^3 - 3x - 15\right)\)[/tex] and [tex]\(\left(x^3 - 3\right)\)[/tex].
1. Set up the division:
We need to divide [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].
2. First term of the quotient:
Divide the leading term of the numerator ([tex]\(x^4\)[/tex]) by the leading term of the denominator ([tex]\(x^3\)[/tex]):
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
So the first term of the quotient is [tex]\(x\)[/tex].
3. Multiply and subtract:
Multiply [tex]\(x\)[/tex] by the entire denominator [tex]\(x^3 - 3\)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
Now, subtract this product from the original numerator:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15
\][/tex]
4. Next term of the quotient:
Now, take the new numerator [tex]\(5x^3 - 15\)[/tex] and divide the leading term [tex]\(5x^3\)[/tex] by the leading term [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
So the next term of the quotient is [tex]\(+5\)[/tex].
5. Multiply and subtract again:
Multiply [tex]\(5\)[/tex] by the entire denominator [tex]\(x^3 - 3\)[/tex]:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
Subtract this product from the current numerator:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
6. Conclusion:
There are no more terms left to bring down, so the division is complete. The quotient of the polynomial division is:
[tex]\[
x + 5
\][/tex]
Therefore, the quotient of [tex]\(\left(x^4 + 5x^3 - 3x - 15\right)\)[/tex] and [tex]\(\left(x^3 - 3\right)\)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
1. Set up the division:
We need to divide [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].
2. First term of the quotient:
Divide the leading term of the numerator ([tex]\(x^4\)[/tex]) by the leading term of the denominator ([tex]\(x^3\)[/tex]):
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
So the first term of the quotient is [tex]\(x\)[/tex].
3. Multiply and subtract:
Multiply [tex]\(x\)[/tex] by the entire denominator [tex]\(x^3 - 3\)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
Now, subtract this product from the original numerator:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15
\][/tex]
4. Next term of the quotient:
Now, take the new numerator [tex]\(5x^3 - 15\)[/tex] and divide the leading term [tex]\(5x^3\)[/tex] by the leading term [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
So the next term of the quotient is [tex]\(+5\)[/tex].
5. Multiply and subtract again:
Multiply [tex]\(5\)[/tex] by the entire denominator [tex]\(x^3 - 3\)[/tex]:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
Subtract this product from the current numerator:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
6. Conclusion:
There are no more terms left to bring down, so the division is complete. The quotient of the polynomial division is:
[tex]\[
x + 5
\][/tex]
Therefore, the quotient of [tex]\(\left(x^4 + 5x^3 - 3x - 15\right)\)[/tex] and [tex]\(\left(x^3 - 3\right)\)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
