If a trinomial has a negative coefficient for the squared term, it is usually easier to factor by first factoring out the common factor - 1. Use this method to factor the following trinomial.
[tex]$-4 x^2-x+5$[/tex]
Select the correct choice below and fill in any answer boxes within your choice.
Answer (2)
Factor out -1: − 4 x 2 − x + 5 = − ( 4 x 2 + x − 5 ) .
Find two numbers that multiply to -20 and add up to 1 (5 and -4).
Rewrite and factor by grouping: 4 x 2 + x − 5 = ( x − 1 ) ( 4 x + 5 ) .
Distribute the negative sign: − ( x − 1 ) ( 4 x + 5 ) = ( 1 − x ) ( 4 x + 5 ) . The final factored form is ( 1 − x ) ( 4 x + 5 ) .
Explanation
Initial Analysis We are asked to factor the trinomial − 4 x 2 − x + 5 . The instruction suggests factoring out a -1 first to make the leading coefficient positive.
Factoring out -1 First, factor out -1 from the trinomial: − 4 x 2 − x + 5 = − ( 4 x 2 + x − 5 )
Finding the factors Now, we need to factor the quadratic 4 x 2 + x − 5 . We look for two numbers that multiply to 4 × ( − 5 ) = − 20 and add up to 1. These numbers are 5 and -4.
Rewriting the middle term Rewrite the middle term using these numbers: 4 x 2 + x − 5 = 4 x 2 + 5 x − 4 x − 5
Factoring by grouping Factor by grouping: 4 x 2 + 5 x − 4 x − 5 = x ( 4 x + 5 ) − 1 ( 4 x + 5 ) = ( x − 1 ) ( 4 x + 5 )
Substituting back Therefore, 4 x 2 + x − 5 = ( x − 1 ) ( 4 x + 5 ) . Substitute this back into the expression with the -1 factored out: − ( 4 x 2 + x − 5 ) = − ( x − 1 ) ( 4 x + 5 )
Distributing the negative sign Distribute the negative sign: − ( x − 1 ) ( 4 x + 5 ) = ( 1 − x ) ( 4 x + 5 ) So the factored form of the trinomial is ( 1 − x ) ( 4 x + 5 ) .
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, ensuring stability and efficiency. Similarly, economists use factoring to model and predict market trends, helping them make informed decisions about investments and resource allocation. By mastering factoring, you're not just learning a math skill; you're developing a powerful tool for problem-solving in various fields.
To factor the trinomial − 4 x 2 − x + 5 , we first factor out -1 to get − ( 4 x 2 + x − 5 ) . Then, we factor the quadratic to find ( x − 1 ) ( 4 x + 5 ) and distribute the negative sign, resulting in the final factored form ( 1 − x ) ( 4 x + 5 ) .
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