Factor the trinomial completely:
[tex]$-16 h^3+80 h^2 y-100 h y^2$[/tex]
Answer (2)
To factor the trinomial − 16 h 3 + 80 h 2 y − 100 h y 2 , first factor out the GCF − 4 h , resulting in − 4 h ( 4 h 2 − 20 h y + 25 y 2 ) . Recognizing the trinomial as a perfect square, we finalize the factorization as − 4 h ( 2 h − 5 y ) 2 .
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Factor out the greatest common factor (GCF): − 4 h ( 4 h 2 − 20 h y + 25 y 2 ) .
Recognize the perfect square trinomial: 4 h 2 − 20 h y + 25 y 2 = ( 2 h − 5 y ) 2 .
Substitute the factored trinomial back into the expression: − 4 h ( 2 h − 5 y ) 2 .
The completely factored form is: − 4 h ( 2 h − 5 y ) 2 .
Explanation
Understanding the Problem We are given the trinomial − 16 h 3 + 80 h 2 y − 100 h y 2 and we want to factor it completely.
Factoring out the GCF First, we look for the greatest common factor (GCF) of the terms. The GCF of − 16 h 3 , 80 h 2 y , and − 100 h y 2 is − 4 h . Factoring out − 4 h from the trinomial, we get: − 4 h ( 4 h 2 − 20 h y + 25 y 2 )
Factoring the Perfect Square Trinomial Now, we examine the trinomial 4 h 2 − 20 h y + 25 y 2 to see if it can be factored further. We notice that it might be a perfect square trinomial. A perfect square trinomial has the form a 2 − 2 ab + b 2 = ( a − b ) 2 or a 2 + 2 ab + b 2 = ( a + b ) 2 .
In our case, we have 4 h 2 = ( 2 h ) 2 and 25 y 2 = ( 5 y ) 2 . Also, the middle term is − 20 h y , which is equal to − 2 ( 2 h ) ( 5 y ) . Therefore, 4 h 2 − 20 h y + 25 y 2 is indeed a perfect square trinomial, and we can write it as ( 2 h − 5 y ) 2 .
Final Factorization Substituting this back into the expression, we have: − 4 h ( 2 h − 5 y ) 2 Thus, the completely factored form of the given trinomial is − 4 h ( 2 h − 5 y ) 2 .
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 systems. Architects use factoring to calculate dimensions and areas when creating building plans. In finance, factoring can be used to analyze investment portfolios and manage risk. Understanding how to factor trinomials allows professionals to solve problems more efficiently and make informed decisions.
