Q. 97 - Precalculus Enhanced with Graphing Utilities

Question

Let $f\left(x\right)=a{x}^2+bx+c$ where $a,b,c$ are odd integers. If x is an integer, show that $f\left(x\right)$ must be an odd integer.

Step-by-Step Solution

Answer

We showed that if $x$ is an integer then $f\left(x\right)$ must be odd

Step 1: Given information

We are given a quadratic equation $f\left(x\right)=a{x}^2+bx+c$ where $a,b,c$are odd.

Step 2: Checking for even or odd

When $x$ is even

We have,

$$\begin{gathered} f\left(x\right)=a{x}^2+bx+c \\ f\left(x\right)=\left(odd\right){\left(even\right)}^2+\left(odd\right)\left(even\right)+\left(odd\right) \\ f\left(x\right)=odd \end{gathered}$$

When x is odd

$$\begin{gathered} f\left(x\right)=a{x}^2+bx+c \\ f\left(x\right)=\left(odd\right){\left(odd\right)}^2+\left(odd\right)\left(odd\right)+\left(odd\right) \\ f\left(x\right)=odd \end{gathered}$$

Step 3: Conclusion

We proved that when $f\left(x\right)=a{x}^2+bx+c$ where $a,b,c$ are odd then if $x$ is a integer then $f\left(x\right)$ is a odd