Find the coefficients $a$ and $b$ of a polynomial which has a double zero $x=1$

Question

Find the coefficients $a$ and $b$ of the polynomial $ax^4+bx^2+1$, if it has a double zero at $x=1$.

How do I begin this because I am out of ideas?

Answer 1

We have $$(ax^4+bx^2+1)_{x=1}=0$$ and $$(ax^4+bx^2+1)'_{x=1}=0$$ and solve the following system: $$a+b+1=0$$ and $$4a+2b=0$$

Answer 2

Hint: A polynomial $p(x)$ has a double zero at $x=1$ iff $p(1)=0$ and $p'(1)=0$.

Answer 3

Shortcut:

Notice that $$(x_1-1)^2=x_1^2-2x_1+1$$ so if you let $x_1=x^2$ then you get $$(x^2-1)^2=\color{red}{(x-1)^2}(x+1)^2=x^4-2x^2+1$$ thus $$(a,b)=(1,-2)$$

Answer 4

Since the polynomial is even, if $x=1$ is a double root, $x=-1$ must also be a double root. Therefore

$$ ax^4 + bx^2 + 1 = (x-1)^2(x+1)^2 $$

Expanding the RHS gives $a = 1, b = -2$