Answer :
To find the quotient of the polynomial division of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] by [tex]\((x^3 - 3)\)[/tex], we can perform polynomial long division.
Here's a step-by-step solution:
1. Setup the Division: We write [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] (the dividend) under the long division symbol, and [tex]\((x^3 - 3)\)[/tex] (the divisor) outside.
2. Divide the Leading Terms: Divide the leading term of the dividend [tex]\(x^4\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex], which gives [tex]\(x\)[/tex].
3. Multiply and Subtract: Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by this result [tex]\(x\)[/tex], giving [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex]. Subtraction:
- [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- [tex]\(- (x^4 - 3x)\)[/tex]
----------------------------------
- Result: [tex]\(5x^3 + 0x^2 - 3x - 15\)[/tex]
4. Repeat the Process: Now, divide the new leading term [tex]\(5x^3\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex], so [tex]\(5x^3 \div x^3 = 5\)[/tex].
5. Multiply and Subtract Again: Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by this result [tex]\(5\)[/tex], giving [tex]\(5 \cdot (x^3 - 3) = 5x^3 - 15\)[/tex]. Subtraction:
- [tex]\(5x^3 + 0x^2 - 3x - 15\)[/tex]
- [tex]\(- (5x^3 - 15)\)[/tex]
----------------------------------
- Result: [tex]\(0x^2 - 3x\)[/tex]
6. Finish Division: Since the degree of the remainder [tex]\((-3x)\)[/tex] is less than the degree of the divisor [tex]\((x^3 - 3)\)[/tex], our division ends here.
The quotient is [tex]\(x + 5\)[/tex] and the remainder is [tex]\(-3x\)[/tex].
Therefore, the quotient of [tex]\(\left(x^4 + 5x^3 - 3x - 15\right)\)[/tex] divided by [tex]\(\left(x^3 - 3\right)\)[/tex] is [tex]\(x + 5\)[/tex]. So the correct answer is:
- [tex]\(x+5\)[/tex]
Here's a step-by-step solution:
1. Setup the Division: We write [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] (the dividend) under the long division symbol, and [tex]\((x^3 - 3)\)[/tex] (the divisor) outside.
2. Divide the Leading Terms: Divide the leading term of the dividend [tex]\(x^4\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex], which gives [tex]\(x\)[/tex].
3. Multiply and Subtract: Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by this result [tex]\(x\)[/tex], giving [tex]\(x \cdot (x^3 - 3) = x^4 - 3x\)[/tex]. Subtraction:
- [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- [tex]\(- (x^4 - 3x)\)[/tex]
----------------------------------
- Result: [tex]\(5x^3 + 0x^2 - 3x - 15\)[/tex]
4. Repeat the Process: Now, divide the new leading term [tex]\(5x^3\)[/tex] by the leading term of the divisor [tex]\(x^3\)[/tex], so [tex]\(5x^3 \div x^3 = 5\)[/tex].
5. Multiply and Subtract Again: Multiply the entire divisor [tex]\((x^3 - 3)\)[/tex] by this result [tex]\(5\)[/tex], giving [tex]\(5 \cdot (x^3 - 3) = 5x^3 - 15\)[/tex]. Subtraction:
- [tex]\(5x^3 + 0x^2 - 3x - 15\)[/tex]
- [tex]\(- (5x^3 - 15)\)[/tex]
----------------------------------
- Result: [tex]\(0x^2 - 3x\)[/tex]
6. Finish Division: Since the degree of the remainder [tex]\((-3x)\)[/tex] is less than the degree of the divisor [tex]\((x^3 - 3)\)[/tex], our division ends here.
The quotient is [tex]\(x + 5\)[/tex] and the remainder is [tex]\(-3x\)[/tex].
Therefore, the quotient of [tex]\(\left(x^4 + 5x^3 - 3x - 15\right)\)[/tex] divided by [tex]\(\left(x^3 - 3\right)\)[/tex] is [tex]\(x + 5\)[/tex]. So the correct answer is:
- [tex]\(x+5\)[/tex]
