Examkin

Question

Difficulty: EasyOperations on Polynomials

A triangle has a base of 2x+42x + 4 inches and a height of x−3x - 3 inches. What is the coefficient of xx when the expression representing the area of the triangle, in square inches, is written in standard form?

Answer: -1

Answer

The coefficient of xx is −1-1.
The area of a triangle is calculated using Area=12×base×height\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}. Substituting the given expressions, we get Area=12(2x+4)(x−3)\text{Area} = \frac{1}{2}(2x + 4)(x - 3). First, we can multiply the 12\frac{1}{2} by (2x+4)(2x + 4), which simplifies to x+2x + 2. Next, we expand (x+2)(x−3)(x + 2)(x - 3) using FOIL to get x2−3x+2x−6x^2 - 3x + 2x - 6. Combining like terms yields x2−x−6x^2 - x - 6. The coefficient of the xx term in this simplified expression is −1-1.

Step-by-Step Solution

1
State the formula for the area of a triangle.
Area=12×base×height\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
To establish the mathematical relationship.
2
Substitute the given values into the formula.
Area=12(2x+4)(x−3)\text{Area} = \frac{1}{2}(2x + 4)(x - 3)
To express the area in terms of the variable xx.
3
Distribute the fraction to the first binomial.
x+2x + 2
Distributing 12\frac{1}{2} to (2x+4)(2x + 4) simplifies the expression before multiplying.
4
Expand the product of the binomials.
x2−3x+2x−6x^2 - 3x + 2x - 6
Using the FOIL method to multiply (x+2)(x + 2) and (x−3)(x - 3).
5
Combine the linear terms.
x2−x−6x^2 - x - 6
To simplify the polynomial and write it in standard form.
6
Identify the coefficient of xx.
−1-1
The coefficient of the xx term in x2−x−6x^2 - x - 6 is −1-1.

Key Concept

Multiplying binomials and applying formulas in geometric contexts
Estimated Time:1m 0s
Rate this question

Topics

Operations on Polynomials

More questions from this topic