Factor the trinomial completely.
$10 m^6 n+9 m^5 n^2+2 m^4 n^3$
Answer (2)
To factor the trinomial 10 m 6 n + 9 m 5 n 2 + 2 m 4 n 3 , we first find the greatest common factor, which is m 4 n . Then, we factor the quadratic trinomial 10 m 2 + 9 mn + 2 n 2 to get the complete factorization: m 4 n ( 2 m + n ) ( 5 m + 2 n ) .
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Find the greatest common factor (GCF) of the terms: m 4 n .
Factor out the GCF: m 4 n ( 10 m 2 + 9 mn + 2 n 2 ) .
Factor the quadratic trinomial: 10 m 2 + 9 mn + 2 n 2 = ( 2 m + n ) ( 5 m + 2 n ) .
Write the complete factorization: m 4 n ( 2 m + n ) ( 5 m + 2 n ) .
Explanation
Understanding the Problem We are asked to factor the trinomial 10 m 6 n + 9 m 5 n 2 + 2 m 4 n 3 completely.
Finding the Greatest Common Factor First, we identify the greatest common factor (GCF) of the terms. The GCF of the coefficients 10, 9, and 2 is 1. The GCF of the variables m 6 , m 5 , m 4 is m 4 . The GCF of the variables n , n 2 , n 3 is n . Therefore, the GCF of the entire expression is m 4 n .
Factoring out the GCF We factor out the GCF from the trinomial: 10 m 6 n + 9 m 5 n 2 + 2 m 4 n 3 = m 4 n ( 10 m 2 + 9 mn + 2 n 2 ) .
Factoring the Quadratic Trinomial Now, we need to factor the quadratic trinomial 10 m 2 + 9 mn + 2 n 2 . We look for two numbers that multiply to 10 × 2 = 20 and add to 9. These numbers are 4 and 5.
Rewriting the Middle Term We rewrite the middle term using the numbers 4 and 5: 10 m 2 + 9 mn + 2 n 2 = 10 m 2 + 5 mn + 4 mn + 2 n 2 .
Factoring by Grouping We factor by grouping: 10 m 2 + 5 mn + 4 mn + 2 n 2 = 5 m ( 2 m + n ) + 2 n ( 2 m + n ) .
Factoring out the Common Binomial We factor out the common binomial: 5 m ( 2 m + n ) + 2 n ( 2 m + n ) = ( 2 m + n ) ( 5 m + 2 n ) .
Complete Factorization Finally, we write the complete factorization by including the GCF: 10 m 6 n + 9 m 5 n 2 + 2 m 4 n 3 = m 4 n ( 2 m + n ) ( 5 m + 2 n ) .
Examples
Factoring trinomials is a fundamental skill in algebra and has practical applications in various fields. For example, engineers use factoring to simplify complex expressions when designing structures or analyzing systems. Imagine you are designing a rectangular garden where the area is represented by the trinomial 10 m 6 n + 9 m 5 n 2 + 2 m 4 n 3 . By factoring this expression, you can determine the possible dimensions (length and width) of the garden in terms of m and n . This allows you to optimize the layout and make informed decisions about the garden's design.
