Factor the binomial $216 s^2-150$ completely.
Answer (2)
The binomial 216 s 2 − 150 can be factored completely as 6 ( 6 s + 5 ) ( 6 s − 5 ) . First, we find the GCD of the coefficients, which is 6, and then factor out the GCD. Finally, we recognize the difference of squares within the parentheses and factor it accordingly.
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Find the greatest common divisor (GCD) of 216 and 150, which is 6.
Factor out the GCD: 216 s 2 − 150 = 6 ( 36 s 2 − 25 ) .
Recognize the difference of squares: 36 s 2 − 25 = ( 6 s ) 2 − ( 5 ) 2 .
Factor the difference of squares: 6 ( 6 s + 5 ) ( 6 s − 5 ) .
The completely factored binomial is 6 ( 6 s + 5 ) ( 6 s − 5 ) .
Explanation
Understanding the Problem We are given the binomial 216 s 2 − 150 and asked to factor it completely. This means we want to express it as a product of simpler expressions.
Finding the Greatest Common Divisor First, we find the greatest common divisor (GCD) of the coefficients 216 and 150. The GCD is the largest number that divides both 216 and 150. We find that GCD ( 216 , 150 ) = 6 .
Factoring out the GCD Next, we factor out the GCD from the binomial: 216 s 2 − 150 = 6 ( 36 s 2 − 25 ) .
Recognizing the Difference of Squares Now, we observe that the expression inside the parentheses, 36 s 2 − 25 , is a difference of squares. We can rewrite it as ( 6 s ) 2 − ( 5 ) 2 . Recall that the difference of squares a 2 − b 2 can be factored as ( a + b ) ( a − b ) .
Factoring the Difference of Squares Applying the difference of squares factorization, we have 36 s 2 − 25 = ( 6 s + 5 ) ( 6 s − 5 ) .
Complete Factorization Finally, we write the complete factorization of the original binomial: 216 s 2 − 150 = 6 ( 6 s + 5 ) ( 6 s − 5 ) .
Examples
Factoring binomials is useful in many areas of mathematics, such as solving equations, simplifying expressions, and analyzing functions. For example, if you have a quadratic equation like 216 s 2 − 150 = 0 , you can use the factored form 6 ( 6 s + 5 ) ( 6 s − 5 ) = 0 to easily find the solutions s = − 6 5 and s = 6 5 . Factoring also helps in simplifying complex algebraic expressions, making them easier to work with.
