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 perform polynomial long division.
### Step-by-Step Solution:
1. Determine the Degrees:
- The degree of the numerator [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] is 4.
- The degree of the denominator [tex]\(x^3 - 3\)[/tex] is 3.
2. Perform Polynomial Long Division:
- First Term: Divide the first term of the numerator [tex]\(x^4\)[/tex] by the first term of the denominator [tex]\(x^3\)[/tex]. This gives [tex]\(x\)[/tex].
- Multiply and Subtract: Multiply [tex]\(x\)[/tex] by the entire divisor [tex]\(x^3 - 3\)[/tex], which gives [tex]\(x^4 - 3x\)[/tex]. Subtract this from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15
\][/tex]
- Repeat: Now divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(5\)[/tex].
- Multiply and Subtract: Multiply [tex]\(5\)[/tex] by [tex]\(x^3 - 3\)[/tex], which gives [tex]\(5x^3 - 15\)[/tex]. Subtract this from the new polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
3. Quotient and Remainder:
- The quotient we derived from this division process is [tex]\(x + 5\)[/tex].
- The remainder is 0, meaning the division is exact.
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].
### Step-by-Step Solution:
1. Determine the Degrees:
- The degree of the numerator [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] is 4.
- The degree of the denominator [tex]\(x^3 - 3\)[/tex] is 3.
2. Perform Polynomial Long Division:
- First Term: Divide the first term of the numerator [tex]\(x^4\)[/tex] by the first term of the denominator [tex]\(x^3\)[/tex]. This gives [tex]\(x\)[/tex].
- Multiply and Subtract: Multiply [tex]\(x\)[/tex] by the entire divisor [tex]\(x^3 - 3\)[/tex], which gives [tex]\(x^4 - 3x\)[/tex]. Subtract this from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15
\][/tex]
- Repeat: Now divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(5\)[/tex].
- Multiply and Subtract: Multiply [tex]\(5\)[/tex] by [tex]\(x^3 - 3\)[/tex], which gives [tex]\(5x^3 - 15\)[/tex]. Subtract this from the new polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
3. Quotient and Remainder:
- The quotient we derived from this division process is [tex]\(x + 5\)[/tex].
- The remainder is 0, meaning the division is exact.
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].
