To find the width of the rectangle, we need to divide the area by the length.
The area of the rectangle is given as \(21x^2 - x - 10\), and the length is given as \(3x + 2\). Therefore, we need to perform the division:
\[ \text{Width} = \frac{\text{Area}}{\text{Length}} = \frac{21x^2 - x - 10}{3x + 2}. \]
We'll perform polynomial long division to divide \(21x^2 - x - 10\) by \(3x + 2\).
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Divide the leading term:
- The leading term of the numerator is \(21x^2\) and the leading term of the denominator is \(3x\).
- Dividing gives: \[ \frac{21x^2}{3x} = 7x. \]
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Multiply and subtract:
- Multiply \(7x\) by \(3x + 2\): \[ 7x(3x + 2) = 21x^2 + 14x. \]
- Now, subtract this from the original polynomial: \[ (21x^2 - x - 10) - (21x^2 + 14x) = -x - 14x - 10 = -15x - 10. \]
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Repeat the process:
- Now, divide the new leading term \(-15x\) by \(3x\): \[ \frac{-15x}{3x} = -5. \]
- Multiply \(-5\) by \(3x + 2\): \[ -5(3x + 2) = -15x - 10. \]
- Now, subtract this from \(-15x - 10\): \[ (-15x - 10) - (-15x - 10) = 0. \]
Since the remainder is \(0\), we find that:
\[ \frac{21x^2 - x - 10}{3x + 2} = 7x - 5. \]
Thus, the width of the rectangle is:
\[ \text{Width} = 7x - 5. \]