Question

For the polynomial function ƒ(x) = −x6 + 3x4 + 4x2, find the zeros. Then determine the multiplicity at each zero and state whether the graph displays the behavior of a touch or a cross at each intercept.

Answer 1

There is a multiple zero at 0 (which means that it touches there), and there are single zeros at -2 and 2 (which means that they cross). There is also 2 imaginary zeros at i and -i.

You can find this by factoring. Start by pulling out the greatest common factor, which in this case is -x^2.

-x^6 + 3x^4 + 4x^2

-x^2(x^4 - 3x^2 - 4)

Now we can factor the inside of the parenthesis. You do this by finding factors of the last number that add up to the middle number.

-x^2(x^4 - 3x^2 - 4)

-x^2(x^2 - 4)(x^2 + 1)

Now we can use the factors of two perfect squares rule to factor the middle parenthesis.

-x^2(x^2 - 4)(x^2 + 1)

-x^2(x - 2)(x + 2)(x^2 + 1)

We would also want to split the term in the front.

-x^2(x - 2)(x + 2)(x^2 + 1)

(x)(-x)(x - 2)(x + 2)(x^2 + 1)

Now we would set each portion equal to 0 and solve.

First root

x = 0 ---> no work needed

Second root

-x = 0 ---> divide by -1

x = 0

Third root

x - 2 = 0

x = 2

Forth root

x + 2 = 0

x = -2

Fifth and Sixth roots

x^2 + 1 = 0

x^2 = -1

x = +/- $\sqrt{-1}$

x = +/- i