Answer :
To solve the problem of finding the quotient of [tex]\( \frac{x^4 + 5x^3 - 3x - 15}{x^3 - 3} \)[/tex], we need to perform polynomial division.
Here's a step-by-step guide on how polynomial division works:
1. Setup: We want to divide the polynomial [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [tex]\( x^3 - 3 \)[/tex].
2. Divide the leading terms: Begin by dividing the leading term of the numerator [tex]\( x^4 \)[/tex] by the leading term of the denominator [tex]\( x^3 \)[/tex]. This gives us [tex]\( x \)[/tex].
3. Multiply and Subtract: Multiply [tex]\( x \)[/tex] by the entire divisor [tex]\( x^3 - 3 \)[/tex], resulting in [tex]\( x^4 - 3x \)[/tex]. Subtract this from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15
\][/tex]
4. Repeat: Notice that the new polynomial [tex]\( 5x^3 - 15 \)[/tex] has a degree of 3, so again divide the leading term [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex], which gives us [tex]\( 5 \)[/tex].
5. Multiply and Subtract: Multiply [tex]\( 5 \)[/tex] by [tex]\( x^3 - 3 \)[/tex] to get [tex]\( 5x^3 - 15 \)[/tex]. Subtract this from [tex]\( 5x^3 - 15 \)[/tex]:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
6. Conclusion: The division is complete since the remainder is 0. The quotient is [tex]\( x + 5 \)[/tex].
Therefore, the quotient of the division is the polynomial [tex]\( x + 5 \)[/tex].
Here's a step-by-step guide on how polynomial division works:
1. Setup: We want to divide the polynomial [tex]\( x^4 + 5x^3 - 3x - 15 \)[/tex] by [tex]\( x^3 - 3 \)[/tex].
2. Divide the leading terms: Begin by dividing the leading term of the numerator [tex]\( x^4 \)[/tex] by the leading term of the denominator [tex]\( x^3 \)[/tex]. This gives us [tex]\( x \)[/tex].
3. Multiply and Subtract: Multiply [tex]\( x \)[/tex] by the entire divisor [tex]\( x^3 - 3 \)[/tex], resulting in [tex]\( x^4 - 3x \)[/tex]. Subtract this from the original polynomial:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15
\][/tex]
4. Repeat: Notice that the new polynomial [tex]\( 5x^3 - 15 \)[/tex] has a degree of 3, so again divide the leading term [tex]\( 5x^3 \)[/tex] by [tex]\( x^3 \)[/tex], which gives us [tex]\( 5 \)[/tex].
5. Multiply and Subtract: Multiply [tex]\( 5 \)[/tex] by [tex]\( x^3 - 3 \)[/tex] to get [tex]\( 5x^3 - 15 \)[/tex]. Subtract this from [tex]\( 5x^3 - 15 \)[/tex]:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
6. Conclusion: The division is complete since the remainder is 0. The quotient is [tex]\( x + 5 \)[/tex].
Therefore, the quotient of the division is the polynomial [tex]\( x + 5 \)[/tex].
