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The area of a rectangular window is (4x^(2)-21 x-18). Both the length and width are polynomials with integer coefficients. What are the dimensions?

The area of a rectangular window is (4x2−21x−18) \left(4 x^{2}-21 x-18\right) . Both the length and width are polynomials with integer coefficients. What are the dimensions?

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Full solution

Q. The area of a rectangular window is (4x2−21x−18) \left(4 x^{2}-21 x-18\right) . Both the length and width are polynomials with integer coefficients. What are the dimensions?
  1. Factor Quadratic Expression: Factor the quadratic expression 4x2−21x−184x^2 - 21x - 18 to find the length and width.
  2. Find Multiplying Numbers: Find two numbers that multiply to (4×−18)=−72(4\times-18) = -72 and add to −21-21.
  3. Write Area as Binomials: The numbers are −24-24 and +3+3 because −24×3=−72-24 \times 3 = -72 and −24+3=−21-24 + 3 = -21.
  4. Factor by Grouping: Write the area as a product of two binomials: (4x2−24x)+(3x−18)(4x^2 - 24x) + (3x - 18).
  5. Factor out Common Binomial: Factor by grouping: 4x(x−6)+3(x−6)4x(x - 6) + 3(x - 6).
  6. Set Window Dimensions: Factor out the common binomial (x−6)(x - 6): (4x+3)(x−6)(4x + 3)(x - 6).
  7. Set Window Dimensions: Factor out the common binomial (x−6)(x - 6): (4x+3)(x−6)(4x + 3)(x - 6).Set the dimensions of the window as length (4x+3)(4x + 3) and width (x−6)(x - 6).

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