Consider the polynomial function 𝑓 of 𝑥 is equal to negative seven 𝑥 to the five plus two 𝑥 to the four plus 𝑥 cubed minus two 𝑥 squared plus two. Evaluate 𝑓 of two and evaluate 𝑓 of negative two.
𝑓 of two means that the variable 𝑥 is taking the value two. We substitute the value two for 𝑥 throughout the polynomial function. 𝑓 of two is equal to negative seven multiplied by two to the power of five plus two multiplied by two to the power of four plus two cubed minus two multiplied by two squared plus two.
Now we need to evaluate this. Two to the power of five is equal to 32. Two to the power of four is equal to 16. Two cubed is eight. And two squared is four. So we have negative seven multiplied by 32 plus two multiplied by 16 plus eight minus two multiplied by four plus two. This is equal to negative 190.
Next, we need to evaluate 𝑓 of negative two. We do this in exactly the same way as we evaluated 𝑓 of two. But we need to be careful with the negatives.
𝑓 of negative two is equal to negative seven multiplied by negative two to the power of five plus two multiplied by negative two to the power of four plus negative two cubed minus two multiplied by negative two squared plus two.
Now be very careful when evaluating these powers of two. For a negative number, the even powers will give positive numbers. But odd powers will give negative numbers.
The difference between 𝑓 of negative two and 𝑓 of two is that the odd powers, five and three, give negative 32 and negative eight rather than 32 and eight. Evaluating this gives 242. 𝑓 of two is negative 190. 𝑓 of negative two is 242.