To factor `x^2+px+q` find two numbers a e b, whose sum is p, a+b=p, and whose product is q, a * b=q. Then
`x^2+px+q=(x+a)(x+b)` Examples:
1) `x^2+x-12`; you may want to write down a list of the different ways of factoring -12 as the product of two numbers:
1, -12; -1, 12; 2, -6; -2, 6; 3, -4; -3, 4;
Those factors are -3 and 4 because: 4 +(– 3) = 1 4 * (-3)= -12
we have: `x^2+x-12=(x+4)(x-3)`
2) `x^2+9x+20`, Those factor are 4 and 5 because
4 + 5 = 9 4 * 5 = 20
then `x^2+9x+20=(x+4)(x+5)`
The Grouping Method
The grouping method can be used to factor polynomials whenever a common factor exists between the groupings. Example: 2ax – bx + 4ay - 2by
The first two terms have a common factor of x and the last two terms have a common factor of 2y.
So we group the first two terms together and group the last two terms together:
x ( 2a – b ) + 2y ( 2a - b ) the two terms have a common factor of (2a – b ); we have:
( 2a – b ) ( x + 2y )
Cube of binomial
Recalling the cube rule of a binomial, we can apply the rule to the inverse determining the two terms that represent the cubes of the two monomials and the other two terms (triple product of the square of the first term for the second and the triple product of the square of second term for the first):
`a^3+3a^2b+3ab^2+b^3=(a+b)^3`
Examples:
1) `x^3-9x^2y^3+27xy^6-27y^9=`
`=x^3+3*x^2*(-3y^3)+3*x*(-3y^3)^2+(-3y^3)^3=(x-3y^3)^3`
1) `8x^6-12x^4y+6x^2y^2-y^3=`
`=(2x^2)^3+3*(2x^2)^2*(-y)+3*2x^2*(-y)^2+(-y)^3=(2x^2-y)^3`
Characteristic Quadrinomials
A polynomial with 4 terms can be formed by a square of a binomial (3 terms) and by the square of a monomial (1 term);
if the two squares have opposite sign, then we can apply the "difference of two squares pattern". Examples:
1) `a^2+ab+b^2-4c^2=(a+b)^2-(2c^2)^2=(a+b+2c)(a+b-2c)`
2) `x^2-y^2+4yz-4z^2=(x)^2-(y^2-4yz+4z^2)=`
`(x)^2-(y-2z)^2=(x+y-2z)(x-y+2z)`