Examkin

Question

Difficulty: HardFactoring Polynomials

The total revenue from selling a batch of electronic components is modeled by the polynomial 15x2−14x−815x^2 - 14x - 8. If the number of units sold is given by the expression (5x+2)(5x + 2), what binomial represents the price per unit?

Answer: 3x-4 / (3x-4) / 3x - 4 / (3x - 4)

Answer

3x - 4
The correct answer is derived by factoring the revenue polynomial 15x2−14x−815x^2 - 14x - 8. Because revenue equals the number of units multiplied by the price per unit, the missing price expression is the binomial that, when multiplied by (5x+2)(5x + 2), yields the given polynomial. By factoring the expression, we find that 15x2−14x−8=(5x+2)(3x−4)15x^2 - 14x - 8 = (5x + 2)(3x - 4), confirming (3x−4)(3x - 4) is the correct binomial.

Step-by-Step Solution

1
Set up the algebraic relationship based on the context.
Total Revenue = (Number of Units) × (Price per Unit), which translates to 15x2−14x−8=(5x+2)×(Price per Unit)15x^2 - 14x - 8 = (5x + 2) \times (\text{Price per Unit}).
This establishes that the price per unit is the missing factor of the given polynomial.
2
Determine the factoring method for the trinomial 15x2−14x−815x^2 - 14x - 8.
Using the ac-method, multiply the leading coefficient (15) by the constant term (-8) to get −120-120.
This is the standard approach for factoring a trinomial where the leading coefficient is greater than 1.
3
Find two integers that multiply to −120-120 and add to the middle coefficient, −14-14.
The required integers are −20-20 and 66, because −20×6=−120-20 \times 6 = -120 and −20+6=−14-20 + 6 = -14.
These integers allow us to correctly split the middle term for factoring by grouping.
4
Rewrite the trinomial and factor by grouping.
15x2−20x+6x−8=5x(3x−4)+2(3x−4)=(5x+2)(3x−4)15x^2 - 20x + 6x - 8 = 5x(3x - 4) + 2(3x - 4) = (5x + 2)(3x - 4).
Grouping extracts the greatest common factor from pairs of terms, revealing the two binomial factors.
5
Identify the missing factor.
Since one factor is given as (5x+2)(5x + 2), the other factor representing the price is (3x−4)(3x - 4).
This completes the solution, satisfying the initial equation.

Key Concept

Factoring Non-Monic Trinomials
Rate this question

Topics

Factoring Polynomials

More questions from this topic