Answer :
To solve this problem, we need to perform polynomial division to find the quotient of [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] divided by [tex]\( x^3 - 3 \)[/tex].
Here's a step-by-step explanation of how polynomial division works in this case:
1. Divide the leading terms: Divide the leading term of the dividend by the leading term of the divisor.
- The leading term of the dividend [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] is [tex]\( x^4 \)[/tex].
- The leading term of the divisor [tex]\( x^3 - 3 \)[/tex] is [tex]\( x^3 \)[/tex].
- Dividing these gives us [tex]\(\frac{x^4}{x^3} = x\)[/tex].
2. Multiply and subtract: Multiply the entire divisor [tex]\( x^3 - 3 \)[/tex] by the result from the first step, which is [tex]\( x \)[/tex], and subtract this from the original dividend.
- [tex]\( x \times (x^3 - 3) = x^4 - 3x \)[/tex].
- Subtract [tex]\( x^4 - 3x \)[/tex] from the original polynomial: [tex]\( (x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15 \)[/tex].
3. Repeat the process: Now, repeat the process with the new polynomial [tex]\( 5x^3 - 15 \)[/tex]:
- Divide the leading term [tex]\( 5x^3 \)[/tex] by the leading term [tex]\( x^3 \)[/tex] to get [tex]\(\frac{5x^3}{x^3} = 5\)[/tex].
- Multiply the whole divisor [tex]\( x^3 - 3 \)[/tex] by 5.
- [tex]\( 5 \times (x^3 - 3) = 5x^3 - 15 \)[/tex].
- Subtract [tex]\( 5x^3 - 15 \)[/tex] from [tex]\( 5x^3 - 15 \)[/tex] to get zero, which leaves no remainder.
The quotient of the division is therefore [tex]\( x + 5 \)[/tex].
We can conclude that the correct answer to the question is the polynomial:
[tex]\[ x + 5 \][/tex]
This was achieved by using polynomial long division, and the remainder was zero, confirming that the division was exact.
Here's a step-by-step explanation of how polynomial division works in this case:
1. Divide the leading terms: Divide the leading term of the dividend by the leading term of the divisor.
- The leading term of the dividend [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] is [tex]\( x^4 \)[/tex].
- The leading term of the divisor [tex]\( x^3 - 3 \)[/tex] is [tex]\( x^3 \)[/tex].
- Dividing these gives us [tex]\(\frac{x^4}{x^3} = x\)[/tex].
2. Multiply and subtract: Multiply the entire divisor [tex]\( x^3 - 3 \)[/tex] by the result from the first step, which is [tex]\( x \)[/tex], and subtract this from the original dividend.
- [tex]\( x \times (x^3 - 3) = x^4 - 3x \)[/tex].
- Subtract [tex]\( x^4 - 3x \)[/tex] from the original polynomial: [tex]\( (x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15 \)[/tex].
3. Repeat the process: Now, repeat the process with the new polynomial [tex]\( 5x^3 - 15 \)[/tex]:
- Divide the leading term [tex]\( 5x^3 \)[/tex] by the leading term [tex]\( x^3 \)[/tex] to get [tex]\(\frac{5x^3}{x^3} = 5\)[/tex].
- Multiply the whole divisor [tex]\( x^3 - 3 \)[/tex] by 5.
- [tex]\( 5 \times (x^3 - 3) = 5x^3 - 15 \)[/tex].
- Subtract [tex]\( 5x^3 - 15 \)[/tex] from [tex]\( 5x^3 - 15 \)[/tex] to get zero, which leaves no remainder.
The quotient of the division is therefore [tex]\( x + 5 \)[/tex].
We can conclude that the correct answer to the question is the polynomial:
[tex]\[ x + 5 \][/tex]
This was achieved by using polynomial long division, and the remainder was zero, confirming that the division was exact.
