Okay, how do you factor a polynomial when there is a number in front of the highest x term.
For example:
how would you factor 2x^2-11x+15
i know the answer is (2x-5)(x-3), right? So you put the 2x on one side and then you put an x on the other. And then you know that 15*2=30, so you find numbers that multiply to equal 30 and add to equal -11 and you get -6 and -5 and then you take have of the -6 to get a -3, and you put it on the x side because x is half of 2x. Is this why (2x-5)(x-3) is the answer? Is that how you do it for every problem when there is a number in front of the x^2?
(2x-5)(x-3) is the answer because
2*1 = 2, -6 + -5 = -11, and -5*-x = 15, the three coefficients of the binomial answer. I don't agree with or understand your reasoning. There is a bit of trial and error in factoring. If nothing seems to work, use the quadratic equation
[-b +/- sqrt (b^2 - 4ac)]/2a for the roots.
can you plz explain how to do it then?
Fidelia, it looks like you are learning a method called decomposition of the middle term.
I will try to explain it in the simplest way I know how
First you multiply the first number by the last, 2*15=30, you did that.
now make a column of pairs of numbers which when multiplied give you +30
They must be either both positive or both negative to get a positive 30, but the middle number is -11, so obviously the pair of numbers are both negative
-2 -15
-3 -10
-5 -6
-15 -1
Which pair adds up to -11?
it is -5 and -6
so now we know that -11x must be split up into -5x and -6x, so....
2x^2 - 11x + 15
=2x^2 - 5x - 6x + 15
Now use common factoring in "pairs"
=x(2x - 5) - 3(2x - 5) , now we see another common factor of (2x-5)
=(2x-5)(x-3)
Now isn't that easy?
Yes, that's the correct method! The decomposition of the middle term is a common technique used to factor a polynomial when there is a number in front of the highest x term. Here's a step-by-step explanation of the process:
Step 1: Multiply the coefficient of the highest x term (2 in this case) by the constant term (15 in this case). This gives you 2 * 15 = 30.
Step 2: Find two numbers that multiply to give you 30 and add to give you the coefficient of the middle term (-11 in this case). In this example, -5 and -6 are the numbers that satisfy these conditions (-5 * -6 = 30 and -5 + -6 = -11).
Step 3: Use these two numbers to split the middle term. Rewrite the polynomial by replacing -11x with -5x - 6x:
2x^2 - 5x - 6x + 15
Step 4: Group the terms and factor out the common factors from each group. In this case, you can factor out x from the first two terms and -3 from the last two terms:
x(2x - 5) - 3(2x - 5)
Step 5: Notice that the expression (2x - 5) can be factored out as a common factor from both terms. Factor it out:
(2x - 5)(x - 3)
And that gives you the factored form of the polynomial: (2x - 5)(x - 3).
This method can be used for any polynomial with a number in front of the x^2 term. It simplifies the process by breaking down a polynomial into smaller, more manageable parts.