Factor polynomials

Question

I am having trouble with these expressions:

$$x^4 - 23x^2 + 1$$ $$2y^2 - 5xy + 2x^2 - ay - ax - a^2$$ $$2x^3 - 4x^2y - x^2z + 2xy^2 - y^2z +2xyz$$

I tried to consult a chapter on factoring in my textbook. It seems to suggest the following:

Check for expressions like $a^n - b^n$ or $(a + b)^n$.

Remember that $x^2 + (a + b)x + ab =(x + a)(x + b)$.

Complete the square and check if any of the above apply. (For example, $x^2 - 5x + 3$ add $\pm (5/2)^2$ to get the expression equivalent to $(a^n + b^n)$.)

Grouping.

Regarding the first expression I tried the third method i.e. I added $(23/2)^2$ hoping to get the expression equivalent to $a^2 - b^2$.

In the second exercise I tried to factor monomial from different expressions to see some sort of pattern. I also tried to complete the square that is to add different expressions like $\pm(5y/2)^2$. Formula itself reminds me of $(a + b + c)^2$. I am pretty sure the result is of the form $(a + b + c) (a - b + c)$ or something like that but I am still unsure how to proceed.

Regarding the third expression I don’t even know where to start. I tried grouping and other methods but I still can’t see a pattern.

Can someone give me a hint? Am I missing some rule? How to solve problems like that? What goes on in your head while you're looking at the expressions like that?

Answer 1

$$x^4-23x^2+1=x^4+2x^2+1-25x^2=(x^2-5x+1)(x^2+5x+1).$$ $$2x^2-5xy+2y^2-ax-ay-a^2=$$ $$=2x^2-5xy+2y^2+\frac{1}{4}(x+y)^2-\frac{1}{4}(x+y)^2-a(x+y)-a^2=$$ $$=\frac{9}{4}(x-y)^2-\left(\frac{1}{2}(x+y)+a\right)^2=$$ $$=\left(\frac{3}{2}(x-y)-\frac{1}{2}(x+y)-a\right)\left(\frac{3}{2}(x-y)+\frac{1}{2}(x+y)+a\right)=$$ $$=(x-2y-a)(2x-y+a).$$ The third is irreducible.

Answer 2

Substitute $$x^2=t$$ in the first one and solve the quadratic, then you can factorize.Hint for the second: $$y-2x-a$$ is one factor. Hint for the last one: One factor is $$(x-y)^2$$