Question

The trinomial x2 + bx – c has factors of (x + m)(x – n), where m, n, and b are positive. What is the relationship between the values of m and n? Explain.

Answer 1

Sample response: The value of m must be greater than the value of n. When you multiply the binomials, the middle term is the result of combining the outside and inside products. So, bx = –nx + mx, or bx = (–n + m)x. This means that b = –n + m. When adding numbers with opposite signs, you subtract their absolute values, and keep the sign of the number having the larger absolute value. Since b is positive, m must have the larger absolute value.

Answer 2

Given trinomial

`f(x)=x^2+bx-c`

If factors are (x+m)(x-n), then the expanded form of this expression must be identical to f(x), i.e.

`(x+m)(x-n)=x^2+(m-n)x-mn`

This means that it is necessary that
m-n=b
mn=c
if (x+m) and (x-n) are factors of f(x).