The area of a rectangle is given by the formula: $\text{Area} = \text{Length} \times \text{Width}.$

Given:

• Area = $4x^2 - 13x - 3$
• Length = $x - 3$

To find the width, we can rearrange the area formula to solve for the width: $\text{Width} = \frac{\text{Area}}{\text{Length}} = \frac{4x^2 - 13x - 3}{x - 3}.$

To divide $4x^2 - 13x - 3$ by $x - 3$, we'll use polynomial division.

1. Polynomial Division:
   • Divide the first term of the numerator by the first term of the denominator: $\frac{4x^2}{x} = 4x.$
   • Multiply $x - 3$ by $4x$: $4x \times (x - 3) = 4x^2 - 12x.$
   • Subtract this product from the original polynomial: $(4x^2 - 13x - 3) - (4x^2 - 12x) = -x - 3.$
   • Now, divide the new first term by the first term of the denominator: $\frac{-x}{x} = -1.$
   • Multiply $x - 3$ by $-1$: $-1 \times (x - 3) = -x + 3.$
   • Subtract this from the previous result: $(-x - 3) - (-x + 3) = -6.$

The division shows that: $4x^2 - 13x - 3 = (x - 3)(4x + 1).$

Therefore, the width of the rectangle is $4x + 1$.

Summary:

• Width: $4x + 1$

Would you like more details or have any questions?

Related Questions:

1. How do you factor a quadratic expression?
2. What is the relationship between area, length, and width in a rectangle?
3. How do you use synthetic division for polynomial division?
4. Can all quadratic expressions be factored?
5. What happens if the divisor is not a factor of the polynomial?

Tip: When dividing polynomials, always double-check by multiplying the quotient by the divisor to ensure it equals the original polynomial.