Question

I need help factoring; it includes finding gcf(if it has one),Binomial and Trinomals
x^3+3x^2-x-3
and
2x^4-128

Answer 1

In order to factor the trinomial, you should first look for a nice and clean factor by which you can divide the trinomial to turn it into a binomial. By the Remainder Theorem, if you plug in a number 'a' to the polynomial and get 0 as a result, then (x-a) is a factor of the polynomial. In the case of this trinomial, we can see quickly that plugging in x=1 returns 0, so we know (x-1) is a factor. You can then divide (x^3+3x^2-x-3)/(x-1) either using synthetic division or standard polynomial division to get x^2+4x+3, which can easily be factored to (x+1)(x+3). Therefore, the factorization for the given trinomial is (x-1)(x+3)(x+1). As for the given binomial, you can start by factoring out a 2 from the expression, yielding 2(x^4-64). From here, you can use the difference of squares factorization to obtain 2(x^2+8)(x^2-8). Because neither of these terms can be simplified further (they are neither differences of squares nor sums or differences of cubes), this is the prime factorization of 2x^4-128. Looking at the factorizations of both of these polynomials, you can see that they share no common factors. Therefore, the GCF between them is just 1.

Answer 2

Thank You very much! I understand it much better!

Answer 3

I give you 2 TIPS to factor by Shortcut these types of polynomials.
1. TIP 1 for Trinomials (ax^2 + bx + c)
If a + b + c = 0, one real roots is 1 and the other is c/a
If a - b + c = 0, one root is -1 and the other is -c/a
2. TIP 2 for Quadrinomials (ax^3 + bx^2 + cx + d)
If a + b + c + d = 0, one real roots is (1)
If a - b + c - d = 0, one real root is (-1).
Get back to your example: (x^3 + 3x^2 - x -3). In this case a -b + c - d = 0, (Tip 2), one root is (-1). The polynomials can be divided by (x + 1).
(x + 1)(x^2 + 2x - 3).
The trinomial (x^2 + 2x -3) can be factored by Tip 2 because a + b + c = 0. One root is 1 and the other is c/a = -3.
Finally the factors are: (x+1)(x -1)(x + 3).
Remember these TIPS. They will save a lot of time for you.

Answer 4

Correction: The trinomial (x^2 + 2x - 3) can be factored by Tip 1 since a + b + c = 0. One roots is (1) and the other is c/a = -3.