A polynomial of degree 4 that has exactly three distinct x-intercepts and whose graph rises to the left and right

Question

Write a polynomial with the given characteristic.

Answer 1

A polynomial of degree 4 has 4 zeroes. These zeroes may correspond with x intercepts. I say may because the zeroes may be complex numbers which have nothing to do with the real number lines. Also, like the equation y=x⁴ , it 4 zeroes, each of them being the number zero but corresponds to only one x intercept. This is called a repeated root.

 

Also, the general direction of the x⁴ graph, in terms of rising to the left and right, depends on the coefficient of the x⁴ term...also called the leading coefficient. If it is positive, it rises to the left and right. If it is negative, it goes down to both the left and right.

 

So we need three distinct roots which means that on of the roots must be a repeated root.

 

So let's say the repeated root is 3 and the other 2 roots are -1 and 5. That gives us the zeroes

x=3, x=3, x=-1 and x=5.......which means

x-3=0, x-3=0, x+1=0 and x-5=0......which means

(x-3)(x-3)(x+1)(x-5)=0......which means

(x²-6x+9)(x²-4x-5)=0......which means

x⁴ -10x³ +28x² -6x -45 =0 ( if I've multiplied correctly...lol)

 

which means one possible function is y=x⁴-10x³+28x²-6x-45.

 

Hope this helped.