Question

Explain the process for factoring each of the following: a. x^2-25 b. 3x^2-12x-15 c. x^3+2x^2+3x+6

Answer 1

a. (x+5)(x-5)

b. 3(x+1)(x-5)

c. (x^2+3)(x+2)

Explanation:

a. x^2-25

The given expression can be factorized using the formula:

`a^(2) -b^(2) =(a+b)(a-b)\\So,\\x^(2) -25\\=(x)^(2)-(5)^(2)\\=(x+5)(x-5)`

b. 3x^2-12x-15

We can see that 3 is common in all terms

=3(x^2-4x-5)

In order to make factors, the constant will be multiplied by the co-efficient of highest degree variable

So,

`3[x^(2) -4x-5]\\=3[x^(2)-5x+x-5]\\=3[x(x-5)+1(x-5)]\\=3(x+1)(x-5)`

c. x^3+2x^2+3x+6

Combining the first and second pair of terms

`x^(3)+2x^(2)+3x+6 \\=[x^(3)+2x^(2)]+[3x+6]\\Taking\ x^(2)\ common\ from\ first\ two\ terms\\=x^(2) (x+2)+3(x+2)\\=(x^(2)+3)(x+2)`