Answer :
To solve the problem of finding the quotient when the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] is divided by [tex]\((x^3 - 3)\)[/tex], we need to perform polynomial division. Let's go through the steps for this division:
1. Set Up the Division: We are dividing the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex].
2. Perform Polynomial Long Division:
- Step 1: Divide the first term of the numerator (which is [tex]\(x^4\)[/tex]) by the first term of the denominator (which is [tex]\(x^3\)[/tex]). This gives us [tex]\(x\)[/tex].
- Step 2: Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex]. This results in [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex].
- Step 3: Subtract this from the original polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] to find the remainder. After subtraction, we get [tex]\((5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15\)[/tex].
- Step 4: Since the degree of the new polynomial [tex]\(5x^3 - 15\)[/tex] is less than the degree of the divisor [tex]\(x^3 - 3\)[/tex], [tex]\((x^3 - 3)\)[/tex], we stop here.
3. Quotient:
- From the division steps, we have determined that the quotient of the division is [tex]\(x + 5\)[/tex].
So, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
1. Set Up the Division: We are dividing the polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex].
2. Perform Polynomial Long Division:
- Step 1: Divide the first term of the numerator (which is [tex]\(x^4\)[/tex]) by the first term of the denominator (which is [tex]\(x^3\)[/tex]). This gives us [tex]\(x\)[/tex].
- Step 2: Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex]. This results in [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex].
- Step 3: Subtract this from the original polynomial [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] to find the remainder. After subtraction, we get [tex]\((5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 - 15\)[/tex].
- Step 4: Since the degree of the new polynomial [tex]\(5x^3 - 15\)[/tex] is less than the degree of the divisor [tex]\(x^3 - 3\)[/tex], [tex]\((x^3 - 3)\)[/tex], we stop here.
3. Quotient:
- From the division steps, we have determined that the quotient of the division is [tex]\(x + 5\)[/tex].
So, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is [tex]\(\boxed{x + 5}\)[/tex].
