Question

Application: A quadratic polynomial is in the form ax2 – bx + c. What do you know about the factors of c which will be used to factor this polynomial?

Answer 1

The general form of a quadratic equation is

$ax^2+bx+c=0$

The product of two factors must be equal to "ac", and the addition of two factors must be equal to the coefficient of x, "b".

If p and q are the two factors of the constant term c, then we have to factor the quadratic equation using p and q as shown below

$(ax\pm p)(x\pm q)=0$

The sign of the factors will be determinated by the sign of b and c. In our problem, the coefficient b is negative and the coefficient c is positive, therefore, the sign for both the factors is negative.

$a(x-p)(x-q)=0$

We know that the product between the factors of the polynomial times the coefficient of the leading term is a factor of c.

$\begin{gathered} a\cdot(-p)\cdot(-q)=c \\ apq=c \end{gathered}$