Decide whether $(6 k+m)(k+5 m)$ or $(6 k+5 m)(k+r$
$6 k^2+31 m k+5 m^2$
Choose the correct form below.
Answer (2)
The correct factorization of the expression 6 k 2 + 31 mk + 5 m 2 is ( 6 k + m ) ( k + 5 m ) , as this expansion matches the original expression. The alternative option, ( 6 k + 5 m ) ( k + m ) , does not yield the same result. Therefore, the answer is (\boxed{(6k + m)(k + 5m)}.
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Expand both possible factorizations: ( 6 k + m ) ( k + 5 m ) and ( 6 k + 5 m ) ( k + m ) .
The expansion of ( 6 k + m ) ( k + 5 m ) is 6 k 2 + 31 mk + 5 m 2 .
The expansion of ( 6 k + 5 m ) ( k + m ) is 6 k 2 + 11 mk + 5 m 2 .
The correct factorization is ( 6 k + m ) ( k + 5 m ) .
Explanation
Problem Analysis We are given the quadratic expression 6 k 2 + 31 mk + 5 m 2 and two possible factorizations: ( 6 k + m ) ( k + 5 m ) and ( 6 k + 5 m ) ( k + m ) . Our goal is to determine which of these factorizations is correct. To do this, we will expand each factorization and compare the result to the given expression.
Expanding the First Factorization Let's expand the first factorization, ( 6 k + m ) ( k + 5 m ) . Using the distributive property (also known as the FOIL method), we have:
( 6 k + m ) ( k + 5 m ) = 6 k ( k ) + 6 k ( 5 m ) + m ( k ) + m ( 5 m ) = 6 k 2 + 30 mk + mk + 5 m 2 = 6 k 2 + 31 mk + 5 m 2 .
Expanding the Second Factorization Now, let's expand the second factorization, ( 6 k + 5 m ) ( k + m ) . Using the distributive property, we have:
( 6 k + 5 m ) ( k + m ) = 6 k ( k ) + 6 k ( m ) + 5 m ( k ) + 5 m ( m ) = 6 k 2 + 6 mk + 5 mk + 5 m 2 = 6 k 2 + 11 mk + 5 m 2 .
Comparison and Conclusion Comparing the expanded forms with the given expression 6 k 2 + 31 mk + 5 m 2 , we see that the first factorization, ( 6 k + m ) ( k + 5 m ) , matches the given expression exactly. The second factorization, ( 6 k + 5 m ) ( k + m ) , results in 6 k 2 + 11 mk + 5 m 2 , which is not the same as the given expression.
Final Answer Therefore, the correct factorization of 6 k 2 + 31 mk + 5 m 2 is ( 6 k + m ) ( k + 5 m ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra. It's used in many real-world applications, such as optimizing areas and volumes, modeling projectile motion, and solving engineering problems. For example, if you want to design a rectangular garden with a specific area and relationship between the length and width, you might need to factor a quadratic expression to find the dimensions.
