Factor the following polynomial completely.
$6 x^2+10 x y-4 y^2=$
Answer (2)
Factor out the greatest common factor: 2 ( 3 x 2 + 5 x y − 2 y 2 ) .
Factor the quadratic trinomial: ( 3 x − y ) ( x + 2 y ) .
Combine the factors: 2 ( 3 x − y ) ( x + 2 y ) .
The completely factored form is: 2 ( 3 x − y ) ( x + 2 y )
Explanation
Problem Analysis We are given the polynomial 6 x 2 + 10 x y − 4 y 2 and asked to factor it completely.
Factoring out the GCF First, we look for the greatest common factor (GCF) of the coefficients. The GCF of 6, 10, and -4 is 2. We factor out the 2: 2 ( 3 x 2 + 5 x y − 2 y 2 ) .
Factoring the Quadratic Expression Now we need to factor the quadratic expression 3 x 2 + 5 x y − 2 y 2 . We are looking for two binomials of the form ( a x + b y ) ( c x + d y ) such that a c = 3 , a d + b c = 5 , and b d = − 2 .
Finding the Correct Factors We can try different combinations of factors to find the correct one. Since 3 is prime, a and c must be 3 and 1 (or 1 and 3). The factors of -2 are -1 and 2, or 1 and -2. Let's try ( 3 x − y ) ( x + 2 y ) : ( 3 x − y ) ( x + 2 y ) = 3 x 2 + 6 x y − x y − 2 y 2 = 3 x 2 + 5 x y − 2 y 2 This is the correct factorization.
Final Factored Form Therefore, the completely factored form of the polynomial is 2 ( 3 x − y ) ( x + 2 y ) .
Examples
Factoring polynomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to simplify equations when designing structures, ensuring stability and efficiency. Similarly, economists use factoring to analyze supply and demand curves, helping them understand market trends and make informed predictions. Factoring also plays a crucial role in computer science, particularly in cryptography, where it is used to create secure encryption algorithms.
To factor the polynomial 6 x 2 + 10 x y − 4 y 2 , first factor out the GCF of 2, resulting in 2 ( 3 x 2 + 5 x y − 2 y 2 ) . Then, factor the trinomial to get 2 ( 3 x − y ) ( x + 2 y ) , giving the final answer. Therefore, the completely factored form is 2 ( 3 x − y ) ( x + 2 y ) .
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