Answer:
Explanation:
To find the expression for the width of the carpet, we need to perform polynomial division. The divisor will be the expression for the length of the carpet, which is x^2 + 5x + 3. The dividend will be the area of the carpet, which is x^4 + 8x^3 + 21x^2 + 24x + 10.
Performing polynomial division, we divide the area by the length:
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x^2 + 5x + 3 | x^4 + 8x^3 + 21x^2 + 24x + 10
To start the division, we look at the highest degree terms: x^4 divided by x^2 gives us x^2. We multiply the divisor by x^2 and subtract it from the dividend:
x^2
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x^2 + 5x + 3 | x^4 + 8x^3 + 21x^2 + 24x + 10
-(x^4 + 5x^3 + 3x^2)
The result of this subtraction is:
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x^2 + 5x + 3 | 3x^3 + 18x^2 + 24x + 10
Now, we continue the division process by dividing the new polynomial, which is 3x^3 + 18x^2 + 24x + 10, by the divisor x^2 + 5x + 3.
We repeat the steps above until we reach the end of the polynomial division. The quotient obtained will be the expression for the width of the carpet, and any remaining terms will be the remainder.
Performing the polynomial division, we get:
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x^2 + 5x + 3 | x^4 + 8x^3 + 21x^2 + 24x + 10
-(x^4 + 5x^3 + 3x^2)
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3x^3 + 18x^2 + 24x + 10
-(3x^3 + 15x^2 + 9x)
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3x^2 + 15x + 10
-(3x^2 + 15x + 9)
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1
At this point, we have a remainder of 1.
Therefore, the expression for the width of the carpet is:
Width = x^2 + 5x + 3 - (3x^2 + 15x + 9)/(x^2 + 5x + 3)
The quotient is 1, and the remainder is 1. The divisor is x^2 + 5x + 3.
So, the expression for the width of the carpet in the desired form is:
Width = 1 + 1/(x^2 + 5x + 3)