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

Answer:

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

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

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

Step-by-step 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

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