Factoring a Polynominal

Question

• When factoring a polynomial in ax2 + bx + c, where a, b, and c are positive real numbers, should the signs in the binomials be positive, negative, or one of each? • Create an example to verify your claim

Answer 1

all positive or all negative if a,b and c are all positive

generally all positive is for the most simple factors and the answer you probably want

(x+1)(x+1)= x^2+2x+1

but so is

(-x-1)(-x-1) = (-1)(x+1)(-1)(x+1) = x^2+2x+1

or

(-1)(-x-1)(x+1) = x^2+2x+1 a mixture of + and - in the binomials

Answer 2

When factoring a polynomial in the form ax² + bx + c, where a, b, and c are positive real numbers, the signs in the binomials should be one of each — meaning one positive and one negative.

Let's create an example to illustrate this:

Consider the polynomial: 2x² − 5x − 3

To factor this, you would look for two numbers that multiply to the product of the leading coefficient (a) and the constant term (c), and add up to the middle coefficient (b).

In this case:

1. Product of a and c: 2 • (-3) = -6
2. Middle coefficient b: -5

The two numbers that meet these criteria are -6 and 1 because

(-6) + 1 = -5 and (-6) • 1 = -6

So, the factored form is 2x² - 5x - 3 = (2x + 1)(x - 3)

In this example, you can see that one binomial has a positive sign (2x + 1) and the other has a negative sign (x − 3), supporting the claim that when factoring a polynomial in ax² + bx + c, where a, b, and c are positive real numbers, the signs in the binomials should be one of each.