Answer :
To find the quotient of the polynomial [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] divided by [tex]\( x^3 - 3 \)[/tex], we need to perform polynomial long division.
1. Set Up the Division:
- Dividend: [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex]
- Divisor: [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]):
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
- The first term of the quotient is [tex]\( x \)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\( x^3 - 3 \)[/tex] by [tex]\( x \)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
- Subtract this from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2
\][/tex]
[tex]\[
= 5x^3 - 15
\][/tex]
4. Repeat the Process:
- Since [tex]\(5x^3\)[/tex] is the new leading term, divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
- Add this to the quotient, making it [tex]\( x + 5 \)[/tex].
5. Multiply and Subtract Again:
- Multiply the divisor by [tex]\( 5 \)[/tex]:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
6. Conclusion:
- The remainder is 0, confirming that the division is exact.
- The quotient of the division 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]\(x + 5\)[/tex].
1. Set Up the Division:
- Dividend: [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex]
- Divisor: [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]):
[tex]\[
\frac{x^4}{x^3} = x
\][/tex]
- The first term of the quotient is [tex]\( x \)[/tex].
3. Multiply and Subtract:
- Multiply the entire divisor [tex]\( x^3 - 3 \)[/tex] by [tex]\( x \)[/tex]:
[tex]\[
x \cdot (x^3 - 3) = x^4 - 3x
\][/tex]
- Subtract this from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2
\][/tex]
[tex]\[
= 5x^3 - 15
\][/tex]
4. Repeat the Process:
- Since [tex]\(5x^3\)[/tex] is the new leading term, divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex]:
[tex]\[
\frac{5x^3}{x^3} = 5
\][/tex]
- Add this to the quotient, making it [tex]\( x + 5 \)[/tex].
5. Multiply and Subtract Again:
- Multiply the divisor by [tex]\( 5 \)[/tex]:
[tex]\[
5 \cdot (x^3 - 3) = 5x^3 - 15
\][/tex]
- Subtract:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
6. Conclusion:
- The remainder is 0, confirming that the division is exact.
- The quotient of the division 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]\(x + 5\)[/tex].
