Problems from Root and factorization of a polynomial

Factorize the following polynomials:

  1. $$x^2+10x+25$$
  2. $$x^4-625$$
  3. $$x^2+x-6$$
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1) We will try to modify the polynomial to have an expression similar to the square of a sum:


2) We effect a change of variable $$x^2=t$$:


And now we can apply the difference of two squares:


The solutions are $$t=25$$ and $$t=-25$$. We undo the change:

$$x^2=25 \Rightarrow \left\{\begin{array}{c} x=5 \\ x=-5 \end{array} \right.$$

The other polynomial $$x^2+25$$ is irreducible.

3) We look for candidates that are roots of the polynomial, specifically the divisors of the independent term (in this case $$6$$):

values: $$1,-1,2,-2,3,-3$$









  1. The root is $$x=-5$$.
  2. The roots are $$x=5$$ and $$x=-5$$.
  3. The roots are $$x=2$$ and $$x=-3$$.
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