factor completely..?

Question

3x^2 - 11 x + 10


The answer is 3x-5, but most show all steps..

Answer 1

(3x - 5)(x - 2)

Use the quadratic formula to find the roots:

x = [ -(-11) +/- {(-11)^2 - (4*3*10)}^(0.5) ] / (2*3)
x = [ 11 +/- (121 - 120)^(0.5) ] / 6
x = (11 +/- 1) / 6

x = 12/6 = 2 and x = 10/6 = 5/3

Then use the roots to get the factors:

x = 2
x - 2 = 0 one factor

x = 5/3
x - 5/3 = 0
3x - 5 = 0 the other factor

You then multiply the two factors to see if it generates the original polynomial. For instance, you would get the same result for factoring 30x^2 - 110x + 300. Once you have the result of that multiplication, look to see if you need to multiply by another factor ("30" in the 30x^2... example) to get the original polynomial. If so, choose one and feed the factor in. Not necessary in this case.

Done! This works for the most amazingly obnoxious looking polynomials. Beastly things just melt when done like this. All other methods fail at being quick, easy, and accurate with anything that hasn't got a simple factoring. It takes moments longer when it IS simple, so look, decide quickly, and use whichever has best promise.

Answer 2

3x^2 - 11x + 10

First do a side sum showing all the factors of 3 and 10
3 x 1 = 3
10 x 1 = 10
5 x 2 = 10

Select two pairs of numbers that will by addition or subtraction bring us to 11: the number are 3,1,5,2

( 3x - 5)(x - 2)

Answer 3

3 * 10 = 30
-6 * -5 multiply to give 30 and add up to give -11

only -6 is divisible by 3 so it factorises to:

(3x - 5)(x - 2)