How do you do these problems ?

Question

Factor the polynomial completely. If the polynomial cannot be factored, write prime.
z^2 - 121


Problem 2

Factor the polynomial completely. If the polynomial cannot be factored, write prime.
125 - t^3

Answer 1

the first is a difference of 2 squares

(z+11)(z-11)

The second is a difference of two cubes

(5-t)(25+5t+t^2)

the key is to recognize what you have. you will get better at it if you do more problems, with a little Patience and practice you will become good at this.

Answer 2

These can be factored, using two factoring rules for perfect squares and cubes:
a^2-b^2=(a-b)(a+b)
a^3-b^3=(a-b)(a^2+ab+b^2)

The first problem z^2-121 follows the first rule, where "z" would model "a" and 121 would model "b", thus take the square root of each to factor:
(z-11)(z+11)

Problem 2 follows the second rule, just rewrite the problem as -t^3+125. We see that it is slightly different from our rule, so we must factor out a negative to get into the form that we need
-(t^3-125), notice that we never changed the problem by factoring.

Finally we are ready to substitute into the rule, where "t "corresponds to "a" and 125 corresponds to "b". Take the cubic root of each and substitute to get:
-(t-5)(t^2+5t+25)

We examine the second term to see if can be further factored and it can't, as there is not two terms that can multiply together to get 25 and add to get 5, so that is our simplest form

Answer 3

z^2 - 121
(z + 11) (z - 11)


prime