Answer :
Sure! Let's find the quotient of the polynomial division [tex]\((x^4 + 5x^3 - 3x - 15) \div (x^3 - 3)\)[/tex] using polynomial long division.
### Step 1: Set it up
First, we write the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and the divisor [tex]\(x^3 - 3\)[/tex]. We need to divide the highest degree term in the dividend by the highest degree term in the divisor.
### Step 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].
[tex]\[ \frac{x^4}{x^3} = x \][/tex]
So, the first term of the quotient is [tex]\(x\)[/tex].
### Step 3: Multiply and subtract
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex].
[tex]\[ x(x^3 - 3) = x^4 - 3x \][/tex]
- Subtract this result from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 15
\][/tex]
### Step 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].
[tex]\[ \frac{5x^3}{x^3} = 5 \][/tex]
So, the next term of the quotient is [tex]\(5\)[/tex].
- Multiply the entire divisor by [tex]\(5\)[/tex]:
[tex]\[ 5(x^3 - 3) = 5x^3 - 15 \][/tex]
- Subtract this from the current dividend:
[tex]\[
(5x^3 + 0x^2 - 15) - (5x^3 - 15) = 0x^2 + 0
\][/tex]
### Conclusion
Since there is no remainder left and no more terms of higher degree than the divisor in the dividend, we can stop here. The quotient of the division is:
[tex]\(x + 5\)[/tex]
The answer is [tex]\(x + 5\)[/tex].
### Step 1: Set it up
First, we write the dividend [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex] and the divisor [tex]\(x^3 - 3\)[/tex]. We need to divide the highest degree term in the dividend by the highest degree term in the divisor.
### Step 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].
[tex]\[ \frac{x^4}{x^3} = x \][/tex]
So, the first term of the quotient is [tex]\(x\)[/tex].
### Step 3: Multiply and subtract
- Multiply the entire divisor [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex].
[tex]\[ x(x^3 - 3) = x^4 - 3x \][/tex]
- Subtract this result from the dividend:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 - 15
\][/tex]
### Step 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].
[tex]\[ \frac{5x^3}{x^3} = 5 \][/tex]
So, the next term of the quotient is [tex]\(5\)[/tex].
- Multiply the entire divisor by [tex]\(5\)[/tex]:
[tex]\[ 5(x^3 - 3) = 5x^3 - 15 \][/tex]
- Subtract this from the current dividend:
[tex]\[
(5x^3 + 0x^2 - 15) - (5x^3 - 15) = 0x^2 + 0
\][/tex]
### Conclusion
Since there is no remainder left and no more terms of higher degree than the divisor in the dividend, we can stop here. The quotient of the division is:
[tex]\(x + 5\)[/tex]
The answer is [tex]\(x + 5\)[/tex].
