# Problems from Root and factorization of a polynomial

Factorize the following polynomials:

1. $$x^2+10x+25$$
2. $$x^4-625$$
3. $$x^2+x-6$$
See development and solution

### Development:

1) We will try to modify the polynomial to have an expression similar to the square of a sum:

$$x^2+10x+25=x^2+2\cdot5x+5^2=(x+5)^2$$

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

$$t^2-625$$

And now we can apply the difference of two squares:

$$t^2-625=(t+25)\cdot(t-25)$$

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

Therefore:

$$p(1)=1^2+1-6=-4$$

$$p(-1)=(-1)^2+(-1)-6=-6$$

$$p(2)=2^2+2-6=0$$

$$p(-2)=(-2)^2+(-2)-6=-4$$

$$p(3)=3^2+3-6=6$$

$$p(-3)=(-3)^2+(-3)-6=0$$

### Solution:

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