Factor the following polynomial completely:
[tex]$96 x^2-6 y^2=$[/tex]
Answer (2)
Factor out the greatest common factor (GCF) from both terms: 96 x 2 − 6 y 2 = 6 ( 16 x 2 − y 2 ) .
Recognize the expression inside the parentheses as a difference of squares: 16 x 2 − y 2 = ( 4 x ) 2 − y 2 .
Apply the difference of squares factorization: 16 x 2 − y 2 = ( 4 x + y ) ( 4 x − y ) .
Combine the GCF with the factored difference of squares to get the final factored form: 6 ( 4 x + y ) ( 4 x − y ) .
Explanation
Understanding the Problem We are asked to factor the polynomial 96 x 2 − 6 y 2 completely. This means we need to find the greatest common factor (GCF) of the coefficients and then see if we can apply any special factoring patterns, such as the difference of squares.
Factoring out the GCF First, we identify the greatest common factor (GCF) of the coefficients 96 and 6. The GCF of 96 and 6 is 6. We factor out 6 from both terms: 96 x 2 − 6 y 2 = 6 ( 16 x 2 − y 2 )
Applying Difference of Squares Now, we look at the expression inside the parentheses: 16 x 2 − y 2 . We recognize that this is a difference of squares, since 16 x 2 = ( 4 x ) 2 and y 2 = y 2 . The difference of squares pattern is a 2 − b 2 = ( a + b ) ( a − b ) . In this case, a = 4 x and b = y . Applying the difference of squares factorization, we get: 16 x 2 − y 2 = ( 4 x + y ) ( 4 x − y )
Final Factored Form Finally, we combine the GCF with the factored difference of squares: 6 ( 16 x 2 − y 2 ) = 6 ( 4 x + y ) ( 4 x − y ) So, the completely factored form of the polynomial is 6 ( 4 x + y ) ( 4 x − y ) .
Examples
Factoring polynomials is a fundamental skill in algebra and has many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures or circuits. Imagine you are designing a rectangular garden and want to know the possible dimensions given a specific area that can be expressed as a polynomial. Factoring the polynomial representing the area can help you find the possible lengths and widths of the garden. This skill is also crucial in fields like physics, where it's used to solve problems involving motion and energy.
The polynomial 96 x 2 − 6 y 2 can be factored completely by first extracting the GCF of 6, resulting in 6 ( 16 x 2 − y 2 ) . This is then recognized as a difference of squares, which factors to ( 4 x + y ) ( 4 x − y ) . The final factored form is 6 ( 4 x + y ) ( 4 x − y ) .
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