Answer :
To find the quotient of the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] when divided by [tex]\((x^3 - 3)\)[/tex], we use polynomial long division.
1. Write the Polynomials:
- Dividend (numerator): [tex]\( x^4 + 5x^3 + 0x^2 - 3x - 15 \)[/tex]
- Divisor: [tex]\( x^3 - 0x^2 - 0x - 3 \)[/tex]
2. Perform the Division:
- First Term: Divide the leading term of the dividend [tex]\(x^4\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex] to get [tex]\(x\)[/tex].
Multiply the entire divisor by [tex]\(x\)[/tex] to get [tex]\(x^4 - 3x\)[/tex].
- Subtract: Subtract [tex]\(x^4 - 3x\)[/tex] from the original dividend:
[tex]\[
(x^4 + 5x^3 + 0x^2 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15
\][/tex]
- Second Term: Divide the leading term [tex]\(5x^3\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex] to get [tex]\(5\)[/tex].
Multiply the entire divisor by [tex]\(5\)[/tex] to get [tex]\(5x^3 - 15\)[/tex].
- Subtract Again: Subtract:
[tex]\[
(5x^3 + 0x^2 + 0x - 15) - (5x^3 - 15) = 0x^3 + 0x^2 + 0x + 0
\][/tex]
3. Identify the Quotient and the Remainder:
- The Division stops here because the degree of the remaining polynomial is less than the degree of the divisor.
- The quotient is [tex]\(x + 5\)[/tex].
- The remainder is 0.
So, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
1. Write the Polynomials:
- Dividend (numerator): [tex]\( x^4 + 5x^3 + 0x^2 - 3x - 15 \)[/tex]
- Divisor: [tex]\( x^3 - 0x^2 - 0x - 3 \)[/tex]
2. Perform the Division:
- First Term: Divide the leading term of the dividend [tex]\(x^4\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex] to get [tex]\(x\)[/tex].
Multiply the entire divisor by [tex]\(x\)[/tex] to get [tex]\(x^4 - 3x\)[/tex].
- Subtract: Subtract [tex]\(x^4 - 3x\)[/tex] from the original dividend:
[tex]\[
(x^4 + 5x^3 + 0x^2 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15
\][/tex]
- Second Term: Divide the leading term [tex]\(5x^3\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex] to get [tex]\(5\)[/tex].
Multiply the entire divisor by [tex]\(5\)[/tex] to get [tex]\(5x^3 - 15\)[/tex].
- Subtract Again: Subtract:
[tex]\[
(5x^3 + 0x^2 + 0x - 15) - (5x^3 - 15) = 0x^3 + 0x^2 + 0x + 0
\][/tex]
3. Identify the Quotient and the Remainder:
- The Division stops here because the degree of the remaining polynomial is less than the degree of the divisor.
- The quotient is [tex]\(x + 5\)[/tex].
- The remainder is 0.
So, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(x + 5\)[/tex].
