Write the function as the product of (x-4), (x-6), and another expression of degree 2 with no real roots.
Example Solution: y=x^4-10x^3+25x^2-10x+24
Practice makes perfect
We want to find a quartic function with the given x-values as its only real zeros.
x=4 and x=6To do so, we can write the function as the product of (x-4), (x-6), and another expression of degree 2 with no real roots — for example, ( x^2+1 ).
y=(x-4)(x-6)( x^2+1 )
Let's simplify the above by using the Distributive Property.
We found that y=x^4-10x^3+25x^2-10x+24 is a quartic function with 4 and 6 as its only real zeros. Note that there are infinitely many solutions for this exercise. We found only one of them.
Alternative Solution
Solving the exercise by using transformations of a graph.
If you like graphs, there is another way of solving this exercise. Consider the parent function y=x^4.
Since the difference between the given zeros is two, we have to find a quartic function where the roots are 2 units apart, and having them centered at 0 is the easiest. Then we can translate the graph over to the correct place using transformations. Let's find a quartic function with zeros at ± 1, because that will add up to 2. c|c
Distance between & Distance between
6 and 4 & 1 and - 1 [0.8em]
6-4=2 & 1- (- 1) =2
We need to translate the parent function (1)^4=1 units down. Therefore, we need to consider the function y=x^4-1.
Finally, 6 and 4 are 5 units to the right of 1 and - 1 respectively.
This means that we need to translate the last function 5 units to the right. Now, we need to consider the function y=(x-5)^4-1.
We can see above that the quartic function with 6 and 4 as its only real zeros is y=(x-5)^4-1. Note that, although this function is different than the one we found using the algebraic method, it is also a correct answer.
