Simple factor of equation

Question

I have this polynomial: $5z^4-12z^3+30z^2-12z+5$

How do I factor it to get the following?: $(5z^2-2z+1)(z^2-2z+5)$

Can someone show me the procedure to perform whenever I encounter with a case like this? Thank you.

Answer 1

You should exploit the symmetry: write the polynomial as $$ z^2\left(5z^2+\frac{5}{z^2}-12z-\frac{12}{z}-30\right) $$ and observe that $$ z^2+\frac{1}{z^2}=\left(z+\frac{1}{z}\right)^{\!2}-2 $$ so you can rewrite the expression as $$ z^2\left(5\left(z+\frac{1}{z}\right)^{\!2}-12\left(z+\frac{1}{z}\right)-40\right) $$ The polynomial $$ 5t^2-12t-40 $$ has roots $$ a=\frac{6+\sqrt{236}}{5},\quad b=\frac{6-\sqrt{236}}{5} $$ so we get $$ 5z^2\left(z+\frac{1}{z}-a\right)\left(z+\frac{1}{z}-b\right) $$ that can be rewritten as $$ 5(z^2-az+1)(z^2-bz+1) $$

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If your polynomial is $$ 5z^4-12z^3\color{red}{+}30z^2-12z+5 $$ the same procedure would give a polynomial in $t$ without real roots. In particular, there is no real root for the polynomial.

In this case you know that if $\alpha$ is a root, also $\alpha^{-1}$ is root. Since a real factorization exists, the roots must have modulus $1$ and if we pair the conjugate pairs, we get a factorization in the form $$ 5z^4-12z^3+30z^2-12z+5= (5z^2+az+b)(bz^2+az+5) $$ (try seeing why). Now it's quite easy to find $a$ and $b$.

Or you can try finding the complex roots. The procedure reduces to the polynomial $$ 5t^2-12t+20 $$ whose roots are $$ \frac{6+8i}{5},\qquad \frac{6-8i}{5} $$ so you just have to solve the equations $$ z+\frac{1}{z}=\frac{6+8i}{5} \qquad z+\frac{1}{z}=\frac{6-8i}{5} $$

Answer 2

(1)
There's no one single procedure which always works.
People usually use the rational root theorem to see if there's an easy-to-find rational root.
If you used it here, you would have found that $-1$ and $+1$ are not roots.

This polynomial has no rational roots.

See also:

factor the polynomial

(2)
Since the polynomial is of degree 4, you can just write your polynomial as...

$5(z^2+az+b)(z^2+cz+d)$

... then open the brackets and see what equations you get for $a,b,c,d$, then try to satisfy them.