Section 6.2: Factoring Trinomials of the Form $x^2+bx+c$ Worksheet 6.2
Recall: Factoring is unmultiplying.
We're going to unmultiply expressions like
A) $x^2+3x+2$
B) $x^2+5x+6$
C) $x^2+2x+4$
Pop Quiz: Which factorization goes with each polynomial above?
( Hint: multiply out each option.)
1) $(x+2)(x+3)$
2) $(x+1)(x+2)$
3) $(x+2)(x+2)$
A Method for Factoring $x^2+bx+c$
We want to change $x^2+bx+c$ into an expression of the form $$(x+r)(x+s).$$
Multiplying this expression out we get $x^2+(r+s)x+rs.$
The Big Deal: To factor $x^2+bx+c$, all you need to do is find two numbers whose
sum is $b$ and whose product is $c$.
Examples: Factor the following trinomials.
$t^2+11t+30$
$q^2-13q+36$
$\phi^2+8\phi-48$
$x^2+2x+12$
Another Common Form: $x^2+bxy+cy^2$
Fact: Expressions of the form $x^2+bxy+cy^2$ factor into $$(x+ry)(x+sy).$$
Multiplying this expression out we get $x^2+(r+s)xy+rsy^2.$
So again, factoring these kinds of expressions amounts to finding two numbers whose
sum is $b$ and whose product is $c$.
Examples
$b^2+9b v+14v^2$
$\beta^2-11\beta r+30r^2$
$s^2-5st-50t^2$
Other Variarions : Completely factor the following polynomials.
$-\gamma^2+10\gamma-21$
$tc^3-17tc^2+70tc$
$sr^3+7r^2 s^2+12rs^3$