Factor a number, variable, or expression out of the trinomial shown below:
[tex]$5 x^2-25 x+10$[/tex]
A. [tex]$2\left(x^2-5 x+5\right)$[/tex]
B. [tex]$5\left(x^2-5 x+2\right)$[/tex]
C. [tex]$5\left(x^2-20 x+5\right)$[/tex]
D. [tex]$5\left(x^2-10 x+2\right)$[/tex]
Answer (1)
Find the greatest common factor (GCF) of the coefficients: The GCF of 5, -25, and 10 is 5.
Factor out the GCF from the trinomial: 5 x 2 − 25 x + 10 = 5 ( x 2 − 5 x + 2 ) .
Verify the result by distributing the GCF back into the parentheses to ensure it matches the original trinomial.
The factored form of the trinomial is 5 ( x 2 − 5 x + 2 ) .
Explanation
Understanding the Problem We are given the trinomial 5 x 2 − 25 x + 10 . Our goal is to factor out the greatest common factor (GCF) from the coefficients of the terms in the trinomial.
Finding the Greatest Common Factor (GCF) The coefficients of the terms are 5, -25, and 10. We need to find the GCF of these numbers. The GCF is the largest number that divides all three coefficients without leaving a remainder.
Listing Factors The factors of 5 are 1 and 5. The factors of 25 are 1, 5, and 25. The factors of 10 are 1, 2, 5, and 10. The greatest common factor of 5, 25, and 10 is 5.
Factoring out the GCF Now, we factor out the GCF, which is 5, from the trinomial: 5 x 2 − 25 x + 10 = 5 ( x 2 − 5 x + 2 ) We divide each term in the trinomial by 5: 5 5 x 2 = x 2 5 − 25 x = − 5 x 5 10 = 2 So, the factored trinomial is 5 ( x 2 − 5 x + 2 ) .
Selecting the Correct Option Comparing our result with the given options: A. 2 ( x 2 − 5 x + 5 ) B. 5 ( x 2 − 5 x + 2 ) C. 5 ( x 2 − 20 x + 5 ) D. 5 ( x 2 − 10 x + 2 ) The correct option is B, which matches our factored trinomial.
Final Answer Therefore, the factored form of the trinomial 5 x 2 − 25 x + 10 is 5 ( x 2 − 5 x + 2 ) .
Examples
Factoring is a fundamental skill in algebra and is used in many real-world applications. For example, if you are designing a rectangular garden and know the area can be represented by the expression 5 x 2 − 25 x + 10 , factoring this expression to 5 ( x 2 − 5 x + 2 ) can help you determine possible dimensions for the garden. The factored form simplifies the expression and can make it easier to find suitable values for x that result in a practical area for the garden. Factoring also helps in simplifying complex equations in physics, engineering, and economics, making it a versatile tool in problem-solving.
