Answer :
To find the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], we can perform polynomial division. Let's break it down step-by-step:
1. Set up the division: We're dividing [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].
2. Divide the leading terms:
- Divide the leading term of the dividend [tex]\(x^4\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex].
- This gives us [tex]\(x\)[/tex].
3. Multiply and subtract:
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by the result from step 2, which is [tex]\(x\)[/tex]. This gives [tex]\(x^4 - 3x\)[/tex].
- Subtract [tex]\(x^4 - 3x\)[/tex] from the original dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex].
4. Calculate subtraction:
- Performing the subtraction, we find:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15
\][/tex]
5. Repeat the process:
- Now, consider [tex]\(5x^3 - 15\)[/tex]. Divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex], which gives you [tex]\(5\)[/tex].
- Multiply [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex] to get [tex]\(5x^3 - 15\)[/tex].
- Subtract [tex]\(5x^3 - 15\)[/tex] from [tex]\(5x^3 - 15\)[/tex], which results in 0, indicating the remainder is zero.
As a result of the division, the quotient 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're dividing [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] by [tex]\(x^3 - 3\)[/tex].
2. Divide the leading terms:
- Divide the leading term of the dividend [tex]\(x^4\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex].
- This gives us [tex]\(x\)[/tex].
3. Multiply and subtract:
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by the result from step 2, which is [tex]\(x\)[/tex]. This gives [tex]\(x^4 - 3x\)[/tex].
- Subtract [tex]\(x^4 - 3x\)[/tex] from the original dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex].
4. Calculate subtraction:
- Performing the subtraction, we find:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15
\][/tex]
5. Repeat the process:
- Now, consider [tex]\(5x^3 - 15\)[/tex]. Divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex], which gives you [tex]\(5\)[/tex].
- Multiply [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex] to get [tex]\(5x^3 - 15\)[/tex].
- Subtract [tex]\(5x^3 - 15\)[/tex] from [tex]\(5x^3 - 15\)[/tex], which results in 0, indicating the remainder is zero.
As a result of the division, the quotient 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].
