The area of a rectangular field is represented by the polynomial x3+x2−14x−24

. The length of the field is represented by the polynomial x−4
. Determine the expression that represents the width of the field.(1 point)
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1 answer

To find the width of the rectangular field, we can use polynomial long division to divide the area polynomial \( x^3 + x^2 - 14x - 24 \) by the length polynomial \( x - 4 \).

We start with the long division:

  1. Divide the leading term of the area polynomial \( x^3 \) by the leading term of the length polynomial \( x \) to get \( x^2 \).

  2. Multiply \( x^2 \) by \( x - 4 \) to get \( x^3 - 4x^2 \).

  3. Subtract this from the original polynomial: \[ (x^3 + x^2 - 14x - 24) - (x^3 - 4x^2) = 5x^2 - 14x - 24 \]

  4. Now repeat the process with \( 5x^2 \):

    • Divide \( 5x^2 \) by \( x \) to get \( 5x \).
    • Multiply \( 5x \) by \( x - 4 \) to get \( 5x^2 - 20x \).
    • Subtract: \[ (5x^2 - 14x - 24) - (5x^2 - 20x) = 6x - 24 \]
  5. Finally, take \( 6x \):

    • Divide \( 6x \) by \( x \) to get \( 6 \).
    • Multiply \( 6 \) by \( x - 4 \) to get \( 6x - 24 \).
    • Subtract: \[ (6x - 24) - (6x - 24) = 0 \]

The result of the division is:

\[ x^2 + 5x + 6 \]

Therefore, the expression that represents the width of the field is:

\[ \boxed{x^2 + 5x + 6} \]