Completely factor the trinomial, if possible.
7x^2+11x+4
Answer (1)
Calculate the discriminant to check if the trinomial is factorable.
Find two numbers whose product is a c and whose sum is b .
Rewrite the middle term using these two numbers and factor by grouping.
The completely factored trinomial is ( 7 x + 4 ) ( x + 1 ) .
Explanation
Understanding the Problem We are given the trinomial 7 x 2 + 11 x + 4 and asked to factor it completely, if possible.
Checking Factorability To factor the trinomial, we look for two binomials of the form ( a x + b ) ( c x + d ) such that when multiplied, they give us the original trinomial. We can check if the trinomial is factorable by calculating the discriminant Δ = b 2 − 4 a c , where a = 7 , b = 11 , and c = 4 .
Calculating the Discriminant We calculate the discriminant: Δ = 1 1 2 − 4 ( 7 ) ( 4 ) = 121 − 112 = 9 . Since Δ = 9 is a perfect square, the trinomial is factorable.
Finding the Right Numbers Now we need to find two numbers whose product is a c = 7 × 4 = 28 and whose sum is b = 11 . The numbers are 4 and 7.
Rewriting the Middle Term We rewrite the middle term using these two numbers: 7 x 2 + 7 x + 4 x + 4 .
Factoring by Grouping Now we factor by grouping: 7 x ( x + 1 ) + 4 ( x + 1 ) .
Factoring Out the Common Factor We factor out the common factor ( x + 1 ) : ( 7 x + 4 ) ( x + 1 ) .
Final Factorization Therefore, the completely factored trinomial is ( 7 x + 4 ) ( x + 1 ) .
Examples
Factoring trinomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures or analyzing circuits. Architects use factoring to calculate dimensions and areas when designing buildings. Financial analysts use factoring to model and predict market trends.
