Factor $6 x^4-5 x^2+12 x^2-10$ by grouping. What is the resulting expression?
$(6 x+5)(x^2-2)$
$(6 x-5)(x^2+2)$
$(6 x^2+5)(x^2-2)$
$(6 x^2-5)(x^2+2)$
Answer (2)
The expression 6 x 4 − 5 x 2 + 12 x 2 − 10 factors to ( 6 x 2 − 5 ) ( x 2 + 2 ) . By grouping and factoring out common terms, we arrive at this result. The correct choice from the options provided is ( 6 x 2 − 5 ) ( x 2 + 2 ) .
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Group the terms: ( 6 x 4 − 5 x 2 ) + ( 12 x 2 − 10 ) .
Factor out the GCF from each group: x 2 ( 6 x 2 − 5 ) + 2 ( 6 x 2 − 5 ) .
Factor out the common binomial factor: ( 6 x 2 − 5 ) ( x 2 + 2 ) .
The resulting expression is ( 6 x 2 − 5 ) ( x 2 + 2 ) .
Explanation
Understanding the Problem We are asked to factor the expression 6 x 4 − 5 x 2 + 12 x 2 − 10 by grouping. This means we want to rearrange and group terms in such a way that we can factor out a common factor from each group, and then factor out a common binomial factor.
Grouping the Terms First, let's group the terms as follows: ( 6 x 4 − 5 x 2 ) + ( 12 x 2 − 10 )
Factoring out the GCF Now, we factor out the greatest common factor (GCF) from each group. From the first group, we can factor out x 2 : x 2 ( 6 x 2 − 5 ) From the second group, we can factor out 2 : 2 ( 6 x 2 − 5 )
Rewriting the Expression Now we rewrite the expression using these factored terms: x 2 ( 6 x 2 − 5 ) + 2 ( 6 x 2 − 5 )
Factoring out the Common Binomial Factor Notice that ( 6 x 2 − 5 ) is a common binomial factor. We can factor this out: ( 6 x 2 − 5 ) ( x 2 + 2 )
Final Answer Thus, the factored expression is ( 6 x 2 − 5 ) ( x 2 + 2 ) . Comparing this with the given options, we see that the correct answer is ( 6 x 2 − 5 ) ( x 2 + 2 ) .
Examples
Factoring by grouping is a useful technique in many areas of mathematics. For example, suppose you are designing a rectangular garden and want to express the area in a factored form to better understand its dimensions. If the area can be represented by an expression like 6 x 4 − 5 x 2 + 12 x 2 − 10 , factoring it into ( 6 x 2 − 5 ) ( x 2 + 2 ) helps you determine possible lengths and widths of the garden in terms of x . This can be particularly useful when x represents a physical dimension or a design parameter.
