$x^2+bx+c$ (যেখানে $c<0$) এই ধরনের রাশির উৎপাদকে বিশ্লেষণ

Factoring Quadratics with $c<0$

Let's take a look at another way to factor trinomials. These will be trinomials with the form $x^2+bx - c$. This method will be a form of factor by grouping. The leading coefficient can be any number, but $C$ should be a negative number.

The first step in factoring a quadratic in a form $x^2+bx - c$ is to find two numbers that multiply to $c$ and add to $b$. $b$ is the middle term. Assume that we have already found numbers $d$ and $e$, where $d \cdot e=c$ and $d+e=b$. Remember that $c$ is negative, so one of either $d$ or $e$ will also be negative.

The polynomial can be rewritten as $x^2 +dx +ex - c$, where one of either $dx$ or $ex$ is a negative term. We can group the terms into $(x^2+dx)+(ex+c)$. This expression has two terms multiplied by the same binomial. We can factor out the binomial to get the factored form of the quadratic.

Example 1:

$$x^2 + x - 12$$

Since $-3 ⋅ 4 = -12$ and $-3 + 4 = 1$, we can rewrite our polynomial:

$$x^2 - 3x + 4x - 12$$

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

$$x(x - 3) + 4(x - 3)$$

Finally, we factor the binomial $x - 3$ 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 $x - 3$):

$$(x + 4)(x - 3)$$

The product of the two binomials, the pair of factors, should equal the original expression, $x^2 + x - 12$. You can use tables or FOIL to multiple binomials.

Example 2:

$$x^2 - 5x - 14$$ $$= x^2 - 7x + 2x - 14$$ $$= x(x - 7) + 2(x - 7)$$ $$= (x + 2)(x - 7)$$

Test Yourself!

It is important to consider what $c<0$ means. When a number is less than $0$ that means it is a negative number. So generally, we are looking at the case when $c$ is a negative number. This means, when we factor $ax^2+bx-c=( ?x {\color{#8925AE}{+}} ? )( ?x {\color{#8925AE}{-}} ? )$. There are two possible factorizations when $c$ is negative.

• When $ax^2{\color{#29A603}{+}}bx-c=( \text{ ? }x {\color{#8925AE}{+}}\text{ bigger number } )( \text{ ? }x {\color{#8925AE}{-}} \text{ smaller number } )$.

• When $ax^2{\color{#29A603}{-}}bx-c=( \text{ ? }x {\color{#8925AE}{-}}\text{ bigger number } )( \text{ ? }x {\color{#8925AE}{+}} \text{ smaller number } )$.

Let's look at a modeling example of the factoring of $x^2+4x-5$.

Now, let's look at a very similar one. $x^2-4x-5$. Notice the differences are in the positive and negative values of the factors and $x$ in the trinomial.

(Video) Factoring Quadratics with Negative c

by Anusha Rahman

In this video, Anusha Rahman covers how to factor quadratic expressions in standard form when $c<0$.

Summary

There are two essential components:

• The quadratic expression must be in standard form [$ax^2+bx+c$]
• $c<0$ (i.e., when $c$ is negative)

We will be using factoring by grouping to solving these types of quadratics.

In standard form, a quadratic expression looks like this: $ax^2+bx+c$. When $c<0$, the quadratic expression will look more like this: $ax^2+bx-c$.

Let's look at some examples.

Example 1: Factor $x^2-3x-10$

When we factor by grouping, we want to find the factors of $-10$ that add to $-3$. After we write out the factors, we determine that the factors we are looking for are: $-5$ and $+2$.

So, we can write:

$$x^2-3x-10$$ $$=x^2 -5x +2x -10$$ $$=x(x-5) +2(x-5)$$ $$=(x-5)(x+2)$$

We always want to double check by multiplying binomials.

Example 2: Factor $x^2+3x-18$

We're going to do something very similar to what we saw in Example 1. We're going to look for factors of $-18$ that sums to $+3$. After we write out all of the factors, we know that the factors we are looking for are: $-3$ and $+6$.

So, we can write:

$$x^2+3x-18$$ $$=x^2-3x+6x-18$$ $$=x(x-3)+6(x-3)$$ $$=(x-3)(x+6)$$

As always, we want to double check by multiplying the two binomials.