Answer :
To find the quotient of the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex], we can perform polynomial long division. Here's how you can do it step-by-step:
1. Set up the division: Write [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] under the division bar and [tex]\((x^3 - 3)\)[/tex] outside the division bar.
2. Divide the leading terms: Divide the first term of the dividend ([tex]\(x^4\)[/tex]) by the first term of the divisor ([tex]\(x^3\)[/tex]). This gives [tex]\(x\)[/tex]. Write [tex]\(x\)[/tex] as the first term of the quotient.
3. Multiply and subtract: Multiply [tex]\(x\)[/tex] by the entire divisor [tex]\((x^3 - 3)\)[/tex], resulting in [tex]\(x^4 - 3x\)[/tex]. Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15
\][/tex]
4. Bring down the next term: Here, we only have [tex]\(5x^3 - 15\)[/tex] left.
5. Repeat the process: Divide the new leading term ([tex]\(5x^3\)[/tex]) by the leading term of the divisor ([tex]\(x^3\)[/tex]), which gives [tex]\(5\)[/tex]. Write [tex]\(5\)[/tex] as the next term of the quotient.
6. Multiply and subtract: Multiply [tex]\(5\)[/tex] by the divisor [tex]\((x^3 - 3)\)[/tex], resulting in [tex]\(5x^3 - 15\)[/tex]. Subtract this from the current expression:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
7. Check the remainder: After subtraction, the remainder is [tex]\(0\)[/tex], which means the division is exact.
The final quotient is [tex]\(x + 5\)[/tex].
Hence, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
1. Set up the division: Write [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] under the division bar and [tex]\((x^3 - 3)\)[/tex] outside the division bar.
2. Divide the leading terms: Divide the first term of the dividend ([tex]\(x^4\)[/tex]) by the first term of the divisor ([tex]\(x^3\)[/tex]). This gives [tex]\(x\)[/tex]. Write [tex]\(x\)[/tex] as the first term of the quotient.
3. Multiply and subtract: Multiply [tex]\(x\)[/tex] by the entire divisor [tex]\((x^3 - 3)\)[/tex], resulting in [tex]\(x^4 - 3x\)[/tex]. Subtract this result from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15
\][/tex]
4. Bring down the next term: Here, we only have [tex]\(5x^3 - 15\)[/tex] left.
5. Repeat the process: Divide the new leading term ([tex]\(5x^3\)[/tex]) by the leading term of the divisor ([tex]\(x^3\)[/tex]), which gives [tex]\(5\)[/tex]. Write [tex]\(5\)[/tex] as the next term of the quotient.
6. Multiply and subtract: Multiply [tex]\(5\)[/tex] by the divisor [tex]\((x^3 - 3)\)[/tex], resulting in [tex]\(5x^3 - 15\)[/tex]. Subtract this from the current expression:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
7. Check the remainder: After subtraction, the remainder is [tex]\(0\)[/tex], which means the division is exact.
The final quotient is [tex]\(x + 5\)[/tex].
Hence, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
