Which of the binomials below is a factor of this trinomial?
[tex]5 x^2+15 x-50[/tex]
A. [tex]x+10[/tex]
B. [tex]x+2[/tex]
C. [tex]x-10[/tex]
D. [tex]x-2[/tex]
Answer (1)
Test each binomial by substituting its root into the trinomial.
For x + 10 , x = − 10 : 5 ( − 10 ) 2 + 15 ( − 10 ) − 50 = 300 e q 0 .
For x + 2 , x = − 2 : 5 ( − 2 ) 2 + 15 ( − 2 ) − 50 = − 60 e q 0 .
For x − 10 , x = 10 : 5 ( 10 ) 2 + 15 ( 10 ) − 50 = 600 e q 0 .
For x − 2 , x = 2 : 5 ( 2 ) 2 + 15 ( 2 ) − 50 = 0 . Therefore, the answer is x − 2 .
Explanation
Understanding the Problem We are given the trinomial 5 x 2 + 15 x − 50 and asked to find which of the given binomials is a factor. A binomial ( x − a ) is a factor of the trinomial if and only if x = a is a root of the trinomial. In other words, if we substitute x = a into the trinomial, the result should be zero. We will test each of the given binomials to see if it is a factor.
Testing Option A A. x + 10 : If x + 10 is a factor, then x = − 10 is a root. Substituting x = − 10 into the trinomial, we get: 5 ( − 10 ) 2 + 15 ( − 10 ) − 50 = 5 ( 100 ) − 150 − 50 = 500 − 150 − 50 = 300 Since the result is not zero, x + 10 is not a factor.
Testing Option B B. x + 2 : If x + 2 is a factor, then x = − 2 is a root. Substituting x = − 2 into the trinomial, we get: 5 ( − 2 ) 2 + 15 ( − 2 ) − 50 = 5 ( 4 ) − 30 − 50 = 20 − 30 − 50 = − 60 Since the result is not zero, x + 2 is not a factor.
Testing Option C C. x − 10 : If x − 10 is a factor, then x = 10 is a root. Substituting x = 10 into the trinomial, we get: 5 ( 10 ) 2 + 15 ( 10 ) − 50 = 5 ( 100 ) + 150 − 50 = 500 + 150 − 50 = 600 Since the result is not zero, x − 10 is not a factor.
Testing Option D D. x − 2 : If x − 2 is a factor, then x = 2 is a root. Substituting x = 2 into the trinomial, we get: 5 ( 2 ) 2 + 15 ( 2 ) − 50 = 5 ( 4 ) + 30 − 50 = 20 + 30 − 50 = 0 Since the result is zero, x − 2 is a factor.
Conclusion Therefore, the binomial x − 2 is a factor of the trinomial 5 x 2 + 15 x − 50 .
Examples
Factoring trinomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to design structures and calculate stresses. Imagine you are designing a rectangular garden with an area represented by the trinomial 5 x 2 + 15 x − 50 . If you know that one side of the garden is ( x − 2 ) , you can use factoring to find the expression for the other side, which would help you determine the dimensions of the garden.
