Answer :
To solve this problem, we need to perform polynomial division. We have two polynomials:
1. The numerator: [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex]
2. The denominator: [tex]\( x^3 - 3 \)[/tex]
We want to divide the numerator by the denominator to find the quotient. Here are the steps to carry out polynomial division:
1. Divide the first term of the numerator by the first term of the denominator:
- The first term of the numerator is [tex]\( x^4 \)[/tex].
- The first term of the denominator is [tex]\( x^3 \)[/tex].
- Dividing these, we get [tex]\( x^4 \div x^3 = x \)[/tex].
2. Multiply the entire denominator by this result:
- Multiply [tex]\( x \)[/tex] by [tex]\( x^3 - 3 \)[/tex], giving [tex]\( x \cdot (x^3 - 3) = x^4 - 3x \)[/tex].
3. Subtract this result from the numerator:
- Subtract [tex]\( x^4 - 3x \)[/tex] from [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex]:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 12x - 15
\][/tex]
4. Repeat the process with the new polynomial (5x^3 - 12x - 15):
- Divide [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex] to get [tex]\( 5 \)[/tex].
- Multiply the denominator by 5: [tex]\( 5 \cdot (x^3 - 3) = 5x^3 - 15 \)[/tex].
- Subtract [tex]\( 5x^3 - 15 \)[/tex] from [tex]\( 5x^3 - 12x - 15 \)[/tex]:
[tex]\[
(5x^3 - 12x - 15) - (5x^3 - 15) = -12x
\][/tex]
5. Repeat if necessary until the degree of the remainder is less than the degree of the denominator:
- In this case, there is no remainder left, as subtraction yielded a remainder of zero.
Therefore, the quotient from dividing [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [tex]\( x^3 - 3 \)[/tex] is [tex]\( x + 5 \)[/tex].
The correct answer to the given question is the polynomial [tex]\( x + 5 \)[/tex].
1. The numerator: [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex]
2. The denominator: [tex]\( x^3 - 3 \)[/tex]
We want to divide the numerator by the denominator to find the quotient. Here are the steps to carry out polynomial division:
1. Divide the first term of the numerator by the first term of the denominator:
- The first term of the numerator is [tex]\( x^4 \)[/tex].
- The first term of the denominator is [tex]\( x^3 \)[/tex].
- Dividing these, we get [tex]\( x^4 \div x^3 = x \)[/tex].
2. Multiply the entire denominator by this result:
- Multiply [tex]\( x \)[/tex] by [tex]\( x^3 - 3 \)[/tex], giving [tex]\( x \cdot (x^3 - 3) = x^4 - 3x \)[/tex].
3. Subtract this result from the numerator:
- Subtract [tex]\( x^4 - 3x \)[/tex] from [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex]:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 12x - 15
\][/tex]
4. Repeat the process with the new polynomial (5x^3 - 12x - 15):
- Divide [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex] to get [tex]\( 5 \)[/tex].
- Multiply the denominator by 5: [tex]\( 5 \cdot (x^3 - 3) = 5x^3 - 15 \)[/tex].
- Subtract [tex]\( 5x^3 - 15 \)[/tex] from [tex]\( 5x^3 - 12x - 15 \)[/tex]:
[tex]\[
(5x^3 - 12x - 15) - (5x^3 - 15) = -12x
\][/tex]
5. Repeat if necessary until the degree of the remainder is less than the degree of the denominator:
- In this case, there is no remainder left, as subtraction yielded a remainder of zero.
Therefore, the quotient from dividing [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [tex]\( x^3 - 3 \)[/tex] is [tex]\( x + 5 \)[/tex].
The correct answer to the given question is the polynomial [tex]\( x + 5 \)[/tex].
