GED Mathematical Reasoning: Factoring Expression III

Standard form for a second degree polynomial

Standard form: ax^2 + bx + c(where a, b and c are just numbers)

Notice that the terms are in decreasing order, beginning with the term containing the highest power of “x.” This is called “standard form” and it is the desired form for factoring using this grouping method.

The sign preceding the term goes with the coefficient. If the sign is addition, the coefficient is positive. If the sign is subtraction, the coefficient is negative.

 

The AC method of factoring

Consider the trinomial x^2 + 3x - 4

The first step of the “AC” method is to identify a, b, and c. In this example, “a” is equal to 1, “b” is equal to 3 and “c” is equal to -4.

The second step of the AC method is to multiply “a” times “c.” In this example, 2 times -12 is equal to -24.

 a \cdot c = 2 \cdot -12 = -24

Step three of the process is to find two numbers whose sum is “b” and whose product is equal to “a” times “c” – the value found in step 2. Begin by considering the pairs of numbers that multiply to be “ac” – then look for a pair whose sum or difference is “b”. Be sure to watch the positive and negative signs!

_____ + _____ = 3
_____ * _____ = -4

pairs of numbers that multiply to be -4

(1,-4) (-1, 4) (2, -2)

The sum of the pair must be 3, therefore we’ll choose (-1, 4)

Step four is to rewrite the original trinomial as a 4 term polynomial. We’ll do so by replacing the middle term with two terms having coefficients equal to the numbers we found in step 3.

x^2 -x + 4x - 4

This leads to Step 5, which is to factor by grouping. If you completed the previous steps, you will find two sets of parenthesis that match. We will proceed with the factor by grouping process by factoring out.

x(x -1) + 4(x - 1)

(x + 4)(x - 1)

 

Example 1

Factor: 2x^2 - 5x - 12

 

Notice that the terms are in decreasing order, beginning with the term containing the highest power of “x.” This is called “standard form”.

Standard form: ax^2 + bx + c(where a, b and c are just numbers)

The first step of the “AC” method is to identify a, b, and c. In this example, “a” is equal to 2, “b” is equal to -5 and “c” is equal to -12.

algeb79

The second step of the AC method is to multiply “a” times “c.” In this example, 2 times -12 is equal to -24.

a \cdot c = 2 \cdot -12 = -24

Step three of the process is to find two numbers whose sum is “b” and whose product is equal to “a” times “c” – the value found in step 2. Be sure to watch the positive and negative signs! We need to find two numbers that add to be -5 and multiply to be -24. Begin by considering the pairs of numbers that multiply to be 24 – look for a pair whose sum or difference is 5.

_____ + _____ = -5
_____ * _____ = -24

24 times 1 is 24, but the sum or difference of the numbers is not 5.

12 times 2 is 24, but again – the sum or difference of the numbers is not 5.

8 times 3 is equal to 24 and the difference between the two numbers IS 5.

Now we’ll choose the positive and negative signs for 8 and 3 so that they fit the parameters of Step 2.

-8 + 3 = -5
-8 /cdot 3 = -24

Step 4 is to rewrite the original trinomial as a 4 term polynomial. We’ll do so by replacing the middle term with two terms having coefficients equal to the numbers we found in step 3.

2x^2 - 5x - 12 = 2x^2 -8x + 3x - 12

Do you see what we did there?  We simply replaced the middle term, -5x, with something that is equivalent. -8x plus 3x is equal to -5x. You may be wondering WHY? Well, now we have a four-term polynomial that we can factor using a strategy that we already know: factor by grouping!

This leads to Step 5, which is to factor by grouping. We can factor 2x out of the first group of two terms, leaving x – 4 in parenthesis.

2x^2 - 8x = 2x(\frac{2x^2}{2x} - \frac{8x}{2x}) = 2x(x - 4)

We can factor positive 3 out of the second group of two terms, leaving x – 4 in parenthesis.

3x - 12 = 3(\frac{3x}{3} - \frac{12}{3}) = 3(x - 4)

Since our two sets of parenthesis match, we will proceed with the factor by grouping process by factoring out x – 4.

2x^2 -8x + 3x - 12 = (x - 4)(2x + 3)

As a final step, we should check our answer by multiplying.

algeb80x \cdot 2x = 2x^2
x \cdot 3 = 3x
-4 \cdot 2x = -8x
-4 \cdot 3 = -12

Since 3x and negative 8x can be combined to be -5x, the resulting expression is:  2 “x” squared minus 5x minus 12, which is equal to the original trinomial and so our answer checks out!

 

Example 2

Factor: x^2 + 2x - 35

 

In example 2, we have a trinomial in standard form. Let’s dive right in and follow the steps to factor this by grouping.

Let’s begin by identifying a, b, and c.

algeb81

“a” is equal to 1, “b” is equal to 2 and “c” is equal to -35.

Next, we’ll multiply “a” times “c.”

a \cdot c = 1 \cdot -35 = -35

For step three, we need to find two numbers that add to be 2 and multiply to be -35. Forget about positive and negative signs for now.

What numbers multiply to be 35 whose sum or difference is 2?

_____ + _____ = 2
_____ * _____ = -35

35 times 1 is 35, but the sum or difference is not 2.

5 times 7 is equal to 35 and the difference between the numbers IS 2.

If we allow the 5 to be negative and the 7 to be positive, we will satisfy the parameters of Step 2.

-5 + 7 = 2
-5 \cdot 7 = -35

So the two numbers whose sum is 2 and whose product is equal to -35 are:  -5 and 7.

Following Step 4, we’ll rewrite the original trinomial with four terms replacing 2x with the terms -5x and 7x. This gives us:

x^2 + 2x - 35 = x^2 - 5x + 7x - 35

By the way, it does not matter the order of these two replacement terms but it is best to order them so that there is a common factor between the first two terms and the last two terms (in preparation for Step 5).

Now, we can complete Step 5 and factor by grouping. We can factor “x” out of the first group of two terms, leaving x – 5 in parenthesis.

x^2 - 5x = x(\frac{x^2}{x} - \frac{5x}{5}) = x(x - 5)

We can factor positive 7 out of the second group of two terms, leaving x – 5 in parenthesis.

7x - 35 = 7(\frac{7x}{7} - \frac{35}{7}) = 7(x - 5)

Since our two sets of parenthesis match, we will proceed with the factor by grouping process by factoring out x – 5.

x^2 - 5x + 7x - 35 = (x - 5)(x + 7)

Left in the second set of parenthesis is x + 7. Therefore the final answer for example one is:

(x - 5)(x + 7) or (x + 7)(x - 5)

As a final step, let’s check our answer by multiplying.

algeb82x \cdot x = x^2
x \cdot 7 = 7x
-5 \cdot x = -5x
-5 \cdot 7 = -35

Since 7x and negative 5x can be combined to be 2x, the resulting expression is:   “x” squared plus 2x minus 35, which is equal to the original trinomial and so our answer checks out!

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