Questions: Factor completely by using the trial-and-error method. Select "Prime" if the polynomial cannot be factored. 10 q^2 + 7 q - 12 = Prime

Transcript text: Factor completely by using the trial-and-error method. Select "Prime" if the polynomial cannot be factored. \[ 10 q^{2}+7 q-12= \] $\square$ Prime

Solution

Solution Steps

To factor the quadratic polynomial $10q^2 + 7q - 12$ using the trial-and-error method, we need to find two binomials whose product gives the original polynomial. We look for pairs of factors of the constant term (-12) and the leading coefficient (10) that combine to give the middle term (7).

Step 1: Identify the Polynomial

We start with the polynomial $10q^2 + 7q - 12$.

Step 2: Factor the Polynomial

Using the trial-and-error method, we find that the polynomial can be factored into two binomials. The factorization is: \[ 10q^2 + 7q - 12 = (2q + 3)(5q - 4) \]

Step 3: Verify the Factorization

To verify, we can expand the factors: \[ (2q + 3)(5q - 4) = 2q \cdot 5q + 2q \cdot (-4) + 3 \cdot 5q + 3 \cdot (-4) = 10q^2 - 8q + 15q - 12 = 10q^2 + 7q - 12 \] This confirms that the factorization is correct.