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]$-9 x^2-x+10$[/tex]
Select the correct choice below and fill in any answer boxes within your choice.
Answer (2)
Factor out -1 from the trinomial: − 9 x 2 − x + 10 = − ( 9 x 2 + x − 10 ) .
Factor the quadratic 9 x 2 + x − 10 by finding two numbers that multiply to -90 and add to 1 (10 and -9).
Rewrite the middle term and factor by grouping: 9 x 2 + 10 x − 9 x − 10 = ( x − 1 ) ( 9 x + 10 ) .
Distribute the -1: − ( x − 1 ) ( 9 x + 10 ) = ( 1 − x ) ( 9 x + 10 ) . The final factored form is ( 1 − x ) ( 9 x + 10 ) .
Explanation
Understanding the Problem We are given the trinomial − 9 x 2 − x + 10 and asked to factor it. The suggestion is to first factor out a -1 from the expression to make the leading coefficient positive.
Factoring out -1 Factoring out -1, we get: − 9 x 2 − x + 10 = − ( 9 x 2 + x − 10 ) Now we need to factor the quadratic 9 x 2 + x − 10 .
Finding the Right Numbers To factor 9 x 2 + x − 10 , we look for two numbers that multiply to 9 × ( − 10 ) = − 90 and add up to 1. These numbers are 10 and -9. So we can rewrite the middle term as 10 x − 9 x :
9 x 2 + x − 10 = 9 x 2 + 10 x − 9 x − 10
Factoring by Grouping Now we factor by grouping: 9 x 2 + 10 x − 9 x − 10 = x ( 9 x + 10 ) − 1 ( 9 x + 10 ) = ( x − 1 ) ( 9 x + 10 ) So, 9 x 2 + x − 10 = ( x − 1 ) ( 9 x + 10 ) .
Final Factorization Substituting this back into the expression with the -1 factored out, we have: − 9 x 2 − x + 10 = − ( x − 1 ) ( 9 x + 10 ) We can distribute the -1 to either factor. Let's distribute it to the first factor: − ( x − 1 ) ( 9 x + 10 ) = ( − x + 1 ) ( 9 x + 10 ) = ( 1 − x ) ( 9 x + 10 ) Alternatively, we could distribute the -1 to the second factor: − ( x − 1 ) ( 9 x + 10 ) = ( x − 1 ) ( − 9 x − 10 ) Both ( 1 − x ) ( 9 x + 10 ) and ( x − 1 ) ( − 9 x − 10 ) are correct factorizations.
Final Answer Therefore, the factored form of the trinomial is ( 1 − x ) ( 9 x + 10 ) .
Examples
Factoring trinomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to analyze the stability of structures, economists use it to model supply and demand curves, and computer scientists use it to design efficient algorithms. Factoring helps simplify complex expressions, making them easier to work with and understand. Imagine you're designing a bridge and need to calculate the load it can bear. Factoring a quadratic equation representing the bridge's stress can help you find the critical points where the stress is highest, ensuring the bridge's safety.
To factor the trinomial − 9 x 2 − x + 10 , we first factor out -1 to get − ( 9 x 2 + x − 10 ) . Then, we factor the quadratic 9 x 2 + x − 10 to find ( x − 1 ) ( 9 x + 10 ) , giving us the final factored forms of ( 1 − x ) ( 9 x + 10 ) or ( x − 1 ) ( − 9 x − 10 ) .
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