Answer :
To divide the polynomial [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [tex]\( x^3 - 3 \)[/tex], we perform polynomial long division. Here are the steps:
1. Setup the division: Write the dividend [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] under the long division bar and the divisor [tex]\( x^3 - 3 \)[/tex] on the outside.
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], which gives [tex]\( x \)[/tex].
3. Multiply and subtract: Multiply [tex]\( x \)[/tex] by the entire divisor [tex]\( x^3 - 3 \)[/tex], which results in [tex]\( x^4 - 3x \)[/tex]. Subtract this product from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15
\][/tex]
4. Repeat the process: Bring down the next terms if necessary. However, simplify [tex]\( 5x^3 - 15 \)[/tex].
5. Divide the new leading term: Divide the new leading term, [tex]\( 5x^3 \)[/tex], by [tex]\( x^3 \)[/tex], which gives [tex]\( +5 \)[/tex].
6. Multiply and subtract again: Multiply [tex]\( 5 \)[/tex] by [tex]\( x^3 - 3 \)[/tex], resulting in [tex]\( 5x^3 - 15 \)[/tex]. Subtract this from the remaining polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
7. Conclusion: Since the remainder is zero, the division is exact, and the quotient is just the terms we derived from our division: [tex]\( x + 5 \)[/tex].
Thus, the quotient of the division is [tex]\( x + 5 \)[/tex]. This matches option [tex]\( \boxed{x+5} \)[/tex].
1. Setup the division: Write the dividend [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] under the long division bar and the divisor [tex]\( x^3 - 3 \)[/tex] on the outside.
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], which gives [tex]\( x \)[/tex].
3. Multiply and subtract: Multiply [tex]\( x \)[/tex] by the entire divisor [tex]\( x^3 - 3 \)[/tex], which results in [tex]\( x^4 - 3x \)[/tex]. Subtract this product from the original dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15
\][/tex]
4. Repeat the process: Bring down the next terms if necessary. However, simplify [tex]\( 5x^3 - 15 \)[/tex].
5. Divide the new leading term: Divide the new leading term, [tex]\( 5x^3 \)[/tex], by [tex]\( x^3 \)[/tex], which gives [tex]\( +5 \)[/tex].
6. Multiply and subtract again: Multiply [tex]\( 5 \)[/tex] by [tex]\( x^3 - 3 \)[/tex], resulting in [tex]\( 5x^3 - 15 \)[/tex]. Subtract this from the remaining polynomial:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
7. Conclusion: Since the remainder is zero, the division is exact, and the quotient is just the terms we derived from our division: [tex]\( x + 5 \)[/tex].
Thus, the quotient of the division is [tex]\( x + 5 \)[/tex]. This matches option [tex]\( \boxed{x+5} \)[/tex].
