how do u factor polynomials?

Answer 1

use the factor theorem

for example x^3+3x^2-13x-15

to start you list all the facotrs of 15 which are +-1, +-3, +-5, and +-15

then you must substitute all these facotrs for x to find which ever is equal to 0

p(1)=(1)^3+3(1)^2-13(1)-15 =-24
p(-1)=(-1)^3+3(-1)^2-13(-1)-15 = 0

so p(-1) is a factor so one of the factor is x+1

then you must divide the polynomial by x+1 to find the remaining factor
in this case (x+1)(x^2+2x-15)

then factor the quadratic you get
(x+1)(x-3)(x+5)

Answer 2

Quadratic (second degree) polynomials are usually fairly simple to do by trial and error. They will consist of two factors (assuming that they're factorable over the set of integers) of the form (ax+b). Pick the two a's so that their product is the coefficient of x^2, and the two b's so that their product equals the constant. For example:

x^2+5x-6

Because the coefficient of x^2 is 1, your factors will look like (x+b)(x+b). Now, just pick b's that multiply together to make -6: either 1 and -6, -1 and 6, 2 and -3, or -2 and 3.

Which pair to choose? Choose the pair that, when each b is multiplied by the a in the OTHER factor, then those two products added together, will total the coefficient of x. In our example, this would be -1 and 6, as (-1*1)+(6*1)=5, which is the coefficient of x. Therefore, x^2+5x-6 = (x-1)(x+6).

For polynomials of degree higher than two, you can use the Factor Theorem, which states that (x-n) is a factor of polynomial p(x) if p(n)=0, and vice versa. This means that if you find a number n that, when plugged in for x, makes the expression equal 0, then one of your factors will be (x-n).

An easy way to do this is by synthetic division. Arrange your polynomial in decreasing order of degree (such as x^3+8x^2+9x-18; notice that the x^3 term comes first, then the x^2, then the x, then the constant). Then write out the coefficients in a row, inserting a 0 for any missing term--in our example, this would look like

1 8 9 -18

To try a number, follow this procedure:

1. Start by bringing down the first number--in this case, the 1.
2. Multiply it by the number you're trying, then add the product to the next number. Let's try -3, for instance: 1*-3=-3, plus 8=5.
3. Take that sum, multiply it by the number you're trying again, and add to the next number, and so on until you've used all the numbers. Don't forget to use the 0's you may have. In our example: 5*-3=-15, plus 9=-6, times -3=18, plus -18=0.
4. If your final result is 0, you've found a root and a factor. In this case, because -3 yields a result of 0, x-(-3), or (x+3), is a factor of x^3+8x^2+9x-18.
5. When you find a factor, the remaining polynomial will be one degree lower, and its coefficients will be the sums you got as you added numbers together--in our example, they'd be 1 (the first one you brought down), 5, and -6, so the remaining polynomial is x^2+5x-6.

I know that's a lot to digest, but I hope it helps.

Answer 3

If you know the roots a,b,c.... and the coeficient of the highest grade term is q you canwrite the polynom as

P(x) = q*(x-a)*(x-b)*(x-c)*......