The trinomial $x^2+b x-c$ 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)
Expand the factored form: ( x + m ) ( x − n ) = x 2 + ( m − n ) x − mn .
Equate coefficients: b = m − n and c = mn .
Since 0"> b > 0 , we have 0"> m − n > 0 .
Therefore, the relationship between m and n is n"> m > n , so n}"> m > n .
Explanation
Understanding the Problem We are given a trinomial x 2 + b x − c that factors into ( x + m ) ( x − n ) , where m , n , and b are positive. Our goal is to find the relationship between m and n .
Expanding the Factored Form First, let's expand the factored form ( x + m ) ( x − n ) .
( x + m ) ( x − n ) = x 2 − n x + m x − mn = x 2 + ( m − n ) x − mn
Equating Coefficients Now, we can equate the coefficients of the trinomial x 2 + b x − c and the expanded form x 2 + ( m − n ) x − mn . This gives us: b = m − n c = mn
Determining the Relationship We are given that b is positive, so 0"> b > 0 . Therefore, we have: 0"> m − n > 0 n"> m > n
Final Answer Thus, the relationship between m and n is that m is greater than n .
Examples
Consider a scenario where you are designing a rectangular garden. The area of the garden is represented by the trinomial x 2 + b x − c , and you want to find the dimensions of the garden in terms of x . Knowing that the factors are ( x + m ) and ( x − n ) , where m and n are positive, and n"> m > n , helps you determine the possible lengths of the sides. For instance, if m = 5 and n = 2 , the sides of the garden would be ( x + 5 ) and ( x − 2 ) , ensuring that the area is correctly represented by the given trinomial. This concept is useful in various design and construction problems where dimensions need to be determined based on area or volume constraints.
