Answer :
Let's solve this problem by performing polynomial long division.
We need to divide the polynomial [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [tex]\( x^3 - 3 \)[/tex].
1. 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]:
[tex]\(\frac{x^4}{x^3} = x\)[/tex].
2. Multiply and subtract.
Multiply the entire divisor [tex]\( x^3 - 3 \)[/tex] by [tex]\( x \)[/tex] to get [tex]\( x^4 - 3x \)[/tex].
Subtract this result from the original polynomial:
[tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 0x - 15\)[/tex].
3. Repeat the process.
Now divide the first term of the new polynomial [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex]:
[tex]\(\frac{5x^3}{x^3} = 5\)[/tex].
4. Multiply and subtract again.
Multiply the divisor [tex]\( x^3 - 3 \)[/tex] by 5 to get [tex]\( 5x^3 - 15 \)[/tex].
Subtract this from the current polynomial:
[tex]\((5x^3 - 0x^2 - 0x - 15) - (5x^3 - 15) = 0x^3 + 0x^2 - 0x + 0\)[/tex].
The division is complete because the remainder is zero. The quotient is [tex]\( x + 5 \)[/tex].
So, the correct answer is [tex]\( x + 5 \)[/tex].
We need to divide the polynomial [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [tex]\( x^3 - 3 \)[/tex].
1. 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]:
[tex]\(\frac{x^4}{x^3} = x\)[/tex].
2. Multiply and subtract.
Multiply the entire divisor [tex]\( x^3 - 3 \)[/tex] by [tex]\( x \)[/tex] to get [tex]\( x^4 - 3x \)[/tex].
Subtract this result from the original polynomial:
[tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 0x - 15\)[/tex].
3. Repeat the process.
Now divide the first term of the new polynomial [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex]:
[tex]\(\frac{5x^3}{x^3} = 5\)[/tex].
4. Multiply and subtract again.
Multiply the divisor [tex]\( x^3 - 3 \)[/tex] by 5 to get [tex]\( 5x^3 - 15 \)[/tex].
Subtract this from the current polynomial:
[tex]\((5x^3 - 0x^2 - 0x - 15) - (5x^3 - 15) = 0x^3 + 0x^2 - 0x + 0\)[/tex].
The division is complete because the remainder is zero. The quotient is [tex]\( x + 5 \)[/tex].
So, the correct answer is [tex]\( x + 5 \)[/tex].
