How Do You Divide Two Polynomials by Factoring and Canceling?

Factoring the numerator and denominator will allow us to find and cancel out common factors
The numerator is 2x²-x-15
The denominator is x²-2x-3
Rewrite the numerator and denominator in factored form to cancel out common factors
If x=3 the denominator would be zero, so x cannot equal 3
3 is called an 'excluded value'

1. The numerator is 2x²-x-15
2. The only way to get 2x² using two linear terms is to multiply 2x by x!
3. +5 and -3 work because +5•(-3) = -15, and 5•x-3•2x=-x!
1. We can FOIL our two binomials together to see if we get the original trinomial back
2. Also check out the last link below to see a cool alternative to the FOIL method!
1. FOIL (2x+5) and (x-3) together to get the numerator 2x²-x-15 back
2. Multiply the first terms, 2x•x, to get 2x²
3. Multiply the outer terms, 2x•-3, to get -6x
4. Multiply the inner terms, 5•x, to get 5x
5. Multiply the last terms, 5•-3, to get -15
6. Simplifying gives us our original numerator back, so we factored correctly!
1. The denominator is x²-2x-3
1. The denominator is x²-2x-3
2. We need to find a pair of numbers that multiply to -3 and add to -2
3. The factors -3 and 1 give us exactly what we need in the original polynomial
4. So we can use those factors to complete the binomals
1. We factored the numerator into (2x+5)(x-3)
2. We factored the denominator into (x+1)(x-3)
1. Right now we have (2x+5)(x-3) being divided by (x+1)(x-3)
2. The two (x-3)'s can cancel out to give us our answer!
1. We can't let the denominator be zero because we're not allowed to divide by zero
2. So we need to exclude 3 from our solution
3. We call 3 an 'excluded value'

Simplify the rational expression 2x2-x-15 divided by x2-2x-3.