Factor [tex]25 x^2-16[/tex]
A) [tex](5 x-4)^2[/tex]
B) [tex](5 x+4)(5 x-4)[/tex]
C) [tex](5 x+4)^2[/tex]
D) [tex](5 x+1)(5 x-1)[/tex]
Answer (1)
Recognize the expression as a difference of squares. Rewrite the expression as ( 5 x ) 2 − ( 4 ) 2 . Apply the difference of squares factorization: ( 5 x ) 2 − ( 4 ) 2 = ( 5 x + 4 ) ( 5 x − 4 ) . The factored form is ( 5 x + 4 ) ( 5 x − 4 ) .
Explanation
Recognizing the Pattern We are asked to factor the expression 25 x 2 − 16 . This looks like a difference of squares, which has the form a 2 − b 2 .
Rewriting the Expression We can rewrite 25 x 2 − 16 as ( 5 x ) 2 − ( 4 ) 2 . Now it perfectly fits the difference of squares pattern, where a = 5 x and b = 4 .
Applying the Difference of Squares The difference of squares factorization is a 2 − b 2 = ( a + b ) ( a − b ) . Applying this to our expression, we get ( 5 x ) 2 − ( 4 ) 2 = ( 5 x + 4 ) ( 5 x − 4 ) .
Selecting the Correct Option Therefore, the factored form of 25 x 2 − 16 is ( 5 x + 4 ) ( 5 x − 4 ) . Comparing this to the given options, we see that it matches option B.
Examples
Factoring the difference of squares is a common technique in algebra and can be applied in various real-life scenarios. For example, imagine you are designing a square garden with an area of 25 x 2 square feet and you want to remove a square section of 16 square feet to create a pathway. Factoring 25 x 2 − 16 into ( 5 x + 4 ) ( 5 x − 4 ) helps you determine the dimensions of the remaining garden area after creating the pathway. This skill is also crucial in physics, engineering, and computer graphics for simplifying equations and solving problems related to areas, volumes, and other geometric properties.
