$x^2+bx+c$ onde $b<0,\c>0$

When tasked with factoring a polynomial of the form $x^2 + bx + c$ (where $b$ is negative), you first must determine what two numbers equal $c$ when multiplied together but equal $b$ when added together. Remember, this is different from $x^2 + bx + c$, when $b$ is positive.

For instance, let's say that we want to factor the polynomial $x^2 + bx + c$ (again, where $b$ is negative), and let's also say that we have found two numbers $d$ and $e$ such that $-d ⋅ -e = c$ and $-d +(-e) = b$.

We then rewrite our polynomial as $x^2 - dx - ex + c$, use parentheses to group $x^2 - dx$ and $ex - c$, and factor out greatest common factors from both those pairs of terms.

This should give us an expression that consists of two terms, both of which are being multiplied by the same binomial. Once we have factored out that binomial, we should be left with an expression consisting of two binomials that are being multiplied together: this is the factored form of our polynomial.

Example 1:

$$x^2 - 3x + 2$$

Since $-2 ⋅ -1 = 2$ and $-2 + (-1) = -3$, we can rewrite our polynomial as follows:

$$x^2 - 2x - 1x + 2$$

Now, we factor out greatest common factors from our polynomial's first two terms and from our polynomial's last two terms:

$$x(x - 2) - 1(x - 2)$$

Finally, we factor the binomial $x - 2$ out of our expression (note that we are only able to do this because our expression consists of two terms that are both being multiplied by the binomial $x - 2$):

$$(x - 1)(x - 2)$$

Example 2:

$$x^2 - 6x + 9$$ $$= x^2 - 3x - 3x + 9$$ $$= x(x - 3) - 3(x - 3)$$ $$= (x - 3)(x - 3)$$ $$= (x - 3)^2$$

This one looks like a perfect square trinomial!

Test Yourself!

Visual Model of Factoring

Take notice of what $b<0$ and $c>0$ mean. This means the coefficient in front of $x$, which we call $b$ is less than $0$, and the constant that stands alone is greater than $0$. In other words, $b$ is negative and $c$ is positive. This means, when we factor $ax^2-bx+c=( ?x {\color{blue}{-}} ? )( ?x {\color{#8925AE}{-}} ? )$. Let's look at a modeling example of the factoring of $x^2-5x+6$.

This pattern for factoring can be used for any quadratic equation that follows the pattern $ax^2-bx+c$.

(Video) Factoring Trinomials

by mathman1024

This video by mathman1024 goes over some examples of factoring quadratics that contain a mix of addition and subtraction.

$\large{\text{Summary}}$

The first quadratic our instructor shows us how to factor is $x^2-10x+21$. As he demonstrates in the video above, in order to begin factoring this quadratic, we must look at the quadratic's first term ($x^2$) and figure out what two terms multiply to it.

Since our main goal in factoring our quadratic is to rewrite the expression as the product of two binomials, and since both of those binomial factors must contain at least one term that consists of one or more variables, our quadratic's first term can only be broken down into the terms $x$ and $x$. We then use those two terms as the first two terms of our quadratic's factors:

$$x^2-10x+21\\ =(x\ \ \ \ \ \ \ )(x\ \ \ \ \ \ \ )$$

Now, we need to figure out what signs our quadratic's factors will contain. We know that the last terms of our quadratic's factors will have to multiply to $21$, a positive number. We also know that multiplying two numbers only yields a positive result when those two numbers are either both positive or both negative. This means that the last terms of our quadratic's factors must be either both positive or both negative.

We determine this by looking to our quadratic's middle term, $-10x$. Since the last terms of our quadratic's factors must add up to the coefficient of our quadratic's middle term, and since adding two positive numbers together will never give us a negative number, it must be the case that our quadratic's factors have last terms that are both negative:

$$x^2-10x+21\\ =(x-\ \ \ \ \ )(x-\ \ \ \ \ )$$

Next, in order to figure out the numeric value of those last terms, we first must list all the pairs of values we can break our quadratic's last term into. In other words, we need to list the factors of $21$:

$$\begin{array}{cc} 1&21\\ \hline 3&7 \end{array}$$

Again, when factoring a quadratic, we must ensure that the last terms of the quadratic's factors add up to the coefficient of its middle term.

So, which of those pairs of factors we listed above adds up to $10$? Yup, it's $3$ and $7$. We now take those two values and use them to complete our quadratic's factors:

$$x^2-10x+21\\ =(x-3)(x-7)$$

After factoring the quadratic $x^2-10x+21$, our instructor goes over some additional factoring examples that require you to pay close attention to sign.