Question

could you help me with the following,I don't understand.

Find the x-intercepts of the polynomial function. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept.

1.f(x) = (x + 1)(x - 3)(x - 1)^2


5.f(x) = x4 - 16x2

7.x4 + 5x3 - 84x2 = 0

Answer 1

5.f(x) = x^4 - 16x^2

7.x^4 + 5x^3 - 84x^2 = 0

Answer 2

4th degree functions can have a maximum of 4 x-intercepts (in general a nth degree function can have at most n x-intercepts)

They tend to look like a W

for 1.
the factors are (x+1)(x-3)(x-1)^2

so there are x-intercepts at -1 and 3.
The factor (x-1)^2 or (x-1)(x-1) tells us that the graph "touches" the x-axis at 1
Whenever you have a squared factor such as (x-a)^2 the curve would touch at a

For 7. factor it this way

x^2(x^2 + 5x - 84)
=x^2(x+12)(x-7)

the graph would touch at the origin, and cut the x-axis at -12 and 7

let me know what you got for #5