Answer :
To find the quotient of [tex]\( \left(x^4 + 5x^3 - 3x - 15\right) \)[/tex] divided by [tex]\( \left(x^3 - 3\right) \)[/tex], we need to perform polynomial long division. Let's go through the steps:
1. 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]
So, the first term of the quotient is [tex]\( x \)[/tex].
2. Multiply [tex]\( x \)[/tex] by the entire divisor [tex]\( x^3 - 3 \)[/tex] to get:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
3. Subtract the result from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0 - 15
\][/tex]
Simplifying gives:
[tex]\[
5x^3 - 15
\][/tex]
4. Divide the new leading term [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
Add [tex]\( 5 \)[/tex] to the quotient.
5. Multiply [tex]\( 5 \)[/tex] by the divisor [tex]\( x^3 - 3 \)[/tex] to get:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
6. Subtract this from the current dividend:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Since we are left with 0 as the remainder, the division is complete. The quotient is:
[tex]\[
x + 5
\][/tex]
Therefore, the quotient of [tex]\(\left(x^4 + 5x^3 - 3x - 15\right)\)[/tex] divided by [tex]\(\left(x^3 - 3\right)\)[/tex] is [tex]\(x + 5\)[/tex].
1. 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]
So, the first term of the quotient is [tex]\( x \)[/tex].
2. Multiply [tex]\( x \)[/tex] by the entire divisor [tex]\( x^3 - 3 \)[/tex] to get:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
3. Subtract the result from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0 - 15
\][/tex]
Simplifying gives:
[tex]\[
5x^3 - 15
\][/tex]
4. Divide the new leading term [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
Add [tex]\( 5 \)[/tex] to the quotient.
5. Multiply [tex]\( 5 \)[/tex] by the divisor [tex]\( x^3 - 3 \)[/tex] to get:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
6. Subtract this from the current dividend:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Since we are left with 0 as the remainder, the division is complete. The quotient is:
[tex]\[
x + 5
\][/tex]
Therefore, the quotient of [tex]\(\left(x^4 + 5x^3 - 3x - 15\right)\)[/tex] divided by [tex]\(\left(x^3 - 3\right)\)[/tex] is [tex]\(x + 5\)[/tex].
