Factor by grouping: [tex]$6 p^2-2 m c+4 p m-3 p c$
Answer (2)
To factor the expression 6 p 2 − 2 m c + 4 p m − 3 p c , rearrange and group the terms, then factor out the greatest common factors from each group. This leads to the factored form ( 2 p − c ) ( 3 p + 2 m ) . Verification by expanding confirms the correctness of the factorization.
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Rearrange the terms: 6 p 2 + 4 p m − 3 p c − 2 m c .
Factor out the GCF from the first two terms: 2 p ( 3 p + 2 m ) .
Factor out the GCF from the last two terms: − c ( 3 p + 2 m ) .
Combine the factored terms: ( 2 p − c ) ( 3 p + 2 m ) .
Explanation
Rearrange Terms Let's factor the expression 6 p 2 − 2 m c + 4 p m − 3 p c by grouping. First, we need to rearrange the terms to make the grouping easier.
Rearrange Expression We can rearrange the terms as follows: 6 p 2 + 4 p m − 3 p c − 2 m c
Factor out GCF Now, we factor out the greatest common factor (GCF) from the first two terms, which are 6 p 2 and 4 p m . The GCF is 2 p . Factoring 2 p out of the first two terms gives us: 2 p ( 3 p + 2 m ) Next, we factor out the GCF from the last two terms, which are − 3 p c and − 2 m c . The GCF is − c . Factoring − c out of the last two terms gives us: − c ( 3 p + 2 m )
Combine Factored Terms Now we have: 2 p ( 3 p + 2 m ) − c ( 3 p + 2 m ) We can see that ( 3 p + 2 m ) is a common factor in both terms. So, we factor out ( 3 p + 2 m ) :
( 2 p − c ) ( 3 p + 2 m )
Verify Factorization To verify the factorization, we expand the result: ( 2 p − c ) ( 3 p + 2 m ) = 2 p ( 3 p ) + 2 p ( 2 m ) − c ( 3 p ) − c ( 2 m ) = 6 p 2 + 4 p m − 3 p c − 2 m c This matches the original expression, so our factorization is correct.
Final Answer Therefore, the factored form of the expression 6 p 2 − 2 m c + 4 p m − 3 p c is: ( 2 p − c ) ( 3 p + 2 m )
Examples
Factoring by grouping is a useful technique in many areas of mathematics, including solving polynomial equations and simplifying algebraic expressions. In real life, factoring can be used in various applications, such as optimizing resource allocation or designing efficient algorithms. For example, if you are designing a rectangular garden with an area represented by a quadratic expression, factoring that expression can help you determine the possible dimensions of the garden.
