Factor completely.
$75 u^2-12 u^4$
Answer (1)
Find the greatest common factor (GCF) of the terms: 3 u 2 .
Factor out the GCF: 3 u 2 ( 25 − 4 u 2 ) .
Recognize the difference of squares: 25 − 4 u 2 = ( 5 ) 2 − ( 2 u ) 2 .
Factor the difference of squares: 3 u 2 ( 5 − 2 u ) ( 5 + 2 u ) . The completely factored expression is 3 u 2 ( 5 − 2 u ) ( 5 + 2 u ) .
Explanation
Understanding the Problem We are asked to factor the expression 75 u 2 − 12 u 4 completely. This means we want to break it down into its simplest multiplicative components.
Finding the Greatest Common Factor First, we look for the greatest common factor (GCF) of the terms 75 u 2 and 12 u 4 . The GCF of the coefficients 75 and 12 is 3, since 75 = 3 × 25 and 12 = 3 × 4 . The GCF of u 2 and u 4 is u 2 . Therefore, the GCF of the entire expression is 3 u 2 .
Factoring out the GCF Next, we factor out the GCF, 3 u 2 , from the original expression: 75 u 2 − 12 u 4 = 3 u 2 ( 25 − 4 u 2 )
Recognizing the Difference of Squares Now, we observe that the expression inside the parentheses, 25 − 4 u 2 , is a difference of squares. We can rewrite it as: 25 − 4 u 2 = ( 5 ) 2 − ( 2 u ) 2
Factoring the Difference of Squares We can factor the difference of squares using the formula a 2 − b 2 = ( a − b ) ( a + b ) . In this case, a = 5 and b = 2 u . Applying the formula, we get: 25 − 4 u 2 = ( 5 − 2 u ) ( 5 + 2 u )
Writing the Completely Factored Expression Finally, we substitute this back into our expression: 3 u 2 ( 25 − 4 u 2 ) = 3 u 2 ( 5 − 2 u ) ( 5 + 2 u ) Thus, the completely factored expression is 3 u 2 ( 5 − 2 u ) ( 5 + 2 u ) .
Examples
Factoring polynomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures or analyzing circuits. Imagine you are designing a rectangular garden and need to determine the dimensions that will give you a specific area. By expressing the area as a polynomial and factoring it, you can find the possible lengths and widths of the garden. This allows for efficient planning and resource allocation.
