Answer :
To find the quotient of the polynomials [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] and [tex]\((x^3 - 3)\)[/tex], we will perform polynomial long division.
### Step-by-step solution:
1. Set up the division: Write the dividend, which is the numerator [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex], and the divisor, which is the denominator [tex]\(x^3 - 3\)[/tex].
2. Divide the first term: Look at the first term of the dividend [tex]\(x^4\)[/tex] and the first term of the divisor [tex]\(x^3\)[/tex]. Divide these to get the first term of the quotient: [tex]\(\frac{x^4}{x^3} = x\)[/tex].
3. Multiply and subtract: Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by this first term of the quotient [tex]\(x\)[/tex]. Then, subtract this result from the original polynomial:
- Multiply: [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex]
- Subtraction: [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x)\)[/tex] results in [tex]\(5x^3 + 0x^2 - 3x - 15\)[/tex].
4. Repeat the division process: Now treat [tex]\(5x^3 + 0x^2 - 3x - 15\)[/tex] as the new dividend.
- Divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(5\)[/tex].
- Multiply and subtract:
- Multiply: [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex]
- Subtract: [tex]\((5x^3 - 3x - 15) - (5x^3 - 15)\)[/tex] results in [tex]\(0x^2 - 3x + 0\)[/tex].
5. Determine the end of division: Since the degree of the new expression [tex]\( -3x\)[/tex] is less than the degree of the divisor [tex]\(x^3 - 3\)[/tex], we stop here.
After performing these steps, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is:
[tex]\[ x + 5 \][/tex]
So, the correct answer is [tex]\(x + 5\)[/tex].
### Step-by-step solution:
1. Set up the division: Write the dividend, which is the numerator [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex], and the divisor, which is the denominator [tex]\(x^3 - 3\)[/tex].
2. Divide the first term: Look at the first term of the dividend [tex]\(x^4\)[/tex] and the first term of the divisor [tex]\(x^3\)[/tex]. Divide these to get the first term of the quotient: [tex]\(\frac{x^4}{x^3} = x\)[/tex].
3. Multiply and subtract: Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by this first term of the quotient [tex]\(x\)[/tex]. Then, subtract this result from the original polynomial:
- Multiply: [tex]\(x(x^3 - 3) = x^4 - 3x\)[/tex]
- Subtraction: [tex]\((x^4 + 5x^3 - 3x - 15) - (x^4 - 3x)\)[/tex] results in [tex]\(5x^3 + 0x^2 - 3x - 15\)[/tex].
4. Repeat the division process: Now treat [tex]\(5x^3 + 0x^2 - 3x - 15\)[/tex] as the new dividend.
- Divide [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex] to get [tex]\(5\)[/tex].
- Multiply and subtract:
- Multiply: [tex]\(5(x^3 - 3) = 5x^3 - 15\)[/tex]
- Subtract: [tex]\((5x^3 - 3x - 15) - (5x^3 - 15)\)[/tex] results in [tex]\(0x^2 - 3x + 0\)[/tex].
5. Determine the end of division: Since the degree of the new expression [tex]\( -3x\)[/tex] is less than the degree of the divisor [tex]\(x^3 - 3\)[/tex], we stop here.
After performing these steps, the quotient of [tex]\((x^4 + 5x^3 - 3x - 15)\)[/tex] divided by [tex]\((x^3 - 3)\)[/tex] is:
[tex]\[ x + 5 \][/tex]
So, the correct answer is [tex]\(x + 5\)[/tex].
