Find the quotient a^2x^3 + (5a^2b+ab)x^2 + (ab^3+5ab^2)x + b^4 divided by ax + b
^ symbolize power so a^2x^3 means a squared times x cubed.
I don't see any shortcuts to doing long division.
Can you get an answer?
To find the quotient of the expression (a^2x^3 + (5a^2b+ab)x^2 + (ab^3+5ab^2)x + b^4) divided by (ax + b), you can use long division.
First, let's set up the long division:
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(ax + b) | a^2x^3 + (5a^2b+ab)x^2 + (ab^3+5ab^2)x + b^4
Step 1: Divide the first term of the dividend (a^2x^3) by the first term of the divisor (ax), which gives a quotient of a and write it above the division line:
a
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(ax + b) | a^2x^3 + (5a^2b+ab)x^2 + (ab^3+5ab^2)x + b^4
Step 2: Multiply the divisor (ax + b) by the quotient (a). This gives you a^2x^2, which you then subtract from the original dividend:
a
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(ax + b) | a^2x^3 + (5a^2b+ab)x^2 + (ab^3+5ab^2)x + b^4
- (a^2x^3 + abx^2)
Step 3: Bring down the next term of the dividend, which is (ab^3 + 5ab^2)x:
a
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(ax + b) | a^2x^3 + (5a^2b+ab)x^2 + (ab^3+5ab^2)x + b^4
- (a^2x^3 + abx^2)
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ab
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(ax + b) | a^2x^3 + (5a^2b+ab)x^2 + (ab^3+5ab^2)x + b^4
- (a^2x^3 + abx^2)
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Step 4: Divide the first term of the remaining polynomial ((ab^3 + 5ab^2)x) by the first term of the divisor (ax), which gives a quotient of b:
a + b
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(ax + b) | a^2x^3 + (5a^2b+ab)x^2 + (ab^3+5ab^2)x + b^4
- (a^2x^3 + abx^2)
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(ab^3 + 5ab^2)x
- (ab^3 + 5ab^2)x
Step 5: Multiply the divisor (ax + b) by the quotient (b). This gives you b^2x, which you then subtract from the remaining polynomial:
a + b
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(ax + b) | a^2x^3 + (5a^2b+ab)x^2 + (ab^3+5ab^2)x + b^4
- (a^2x^3 + abx^2)
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(ab^3 + 5ab^2)x
- (ab^3 + 5ab^2)x
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b^2x + b^4
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(ax + b) | a^2x^3 + (5a^2b+ab)x^2 + (ab^3+5ab^2)x + b^4
- (a^2x^3 + abx^2)
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(ab^3 + 5ab^2)x
- (ab^3 + 5ab^2)x
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Step 6: Finally, no more terms can be brought down. The remaining polynomial is b^2x + b^4. Since there are no more terms to divide, this is the remainder.
Therefore, the quotient when (a^2x^3 + (5a^2b+ab)x^2 + (ab^3+5ab^2)x + b^4) is divided by (ax + b) is a + b with a remainder of (b^2x + b^4).