Answer :
To find the quotient of the division of the polynomial [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex], we will perform polynomial long division. Here is the step-by-step process:
1. Setup the division:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
2. Divide the first terms:
- Divide [tex]\(x^4\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(x\)[/tex].
3. Multiply and subtract:
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex]. This gives [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract the result [tex]\(x^4 - 3x\)[/tex] from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 3x + 3x - 15
\][/tex]
Simplify: [tex]\(5x^3 - 15\)[/tex].
4. Repeat for the new polynomial:
- Divide the first term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(5\)[/tex].
- Multiply the divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex], giving [tex]\(5x^3 - 15\)[/tex].
- Subtract:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Since the remainder is [tex]\(0\)[/tex] and we've fully divided the polynomial, the 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 [tex]\(\boxed{x+5}\)[/tex].
1. Setup the division:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
2. Divide the first terms:
- Divide [tex]\(x^4\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(x\)[/tex].
3. Multiply and subtract:
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex]. This gives [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex].
- Subtract the result [tex]\(x^4 - 3x\)[/tex] from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 3x + 3x - 15
\][/tex]
Simplify: [tex]\(5x^3 - 15\)[/tex].
4. Repeat for the new polynomial:
- Divide the first term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex], which gives [tex]\(5\)[/tex].
- Multiply the divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex], giving [tex]\(5x^3 - 15\)[/tex].
- Subtract:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Since the remainder is [tex]\(0\)[/tex] and we've fully divided the polynomial, the 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 [tex]\(\boxed{x+5}\)[/tex].
