We will walk through some examples to show you how to factor polynomials.

1. Factor 12y² – 5y

12y² – 5y = y(12y-5)

2. Factor x²y³ + xy

x²y³ + xy = xy (xy²+1)

3. Factor x(x – 2) + 3(2 – x)

x(x – 2) + 3(2 – x) = x(x – 2) – 3(x – 2) = (x – 2)(x – 3)

4. Factor xy – 5y – 2x + 10.

xy – 5y – 2x + 10 = y(x – 5) – 2x + 10 = y(x – 5) – 2(x – 5)

5. Factor x² – 4x + 6x – 24.

x² – 4x + 6x – 24 = x(x – 4) + 6(x – 4) = (x – 4)(x + 6)

6. Factor 6x² – 13x + 6.

6x² – 13x + 6 = 6x² – 9x – 4x + 6 = 3x(2x – 3) – 2(2x – 3) = (2x – 3)(3x – 2)

7. Factor quadratic polynomial x² + ax + b

According to Vieta's Formula, if we have a quadratic x² + ax + b = 0 with solutions p and q, then we know that we can factor it as x² + ax + b = (x-p)(x-q), Note that the first term is x², not ax².  Using the distributive property to expand the right side we get

x² + ax + b = x² - (p+q)x + pq

From above, we can easily factor quadratic polynomial x² + ax + b by finding p and q so that:

p+q = a and

pq = b

Example: Factor x² – 5x + 6.

x² – 5x + 6 = (x – 2)(x – 3), because -2*-3 = 6, and -2+(-3) = -5