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Related Lessons: Factoring Trinomial: Hard Case Factoring Trinomial: "Box" Method

 

Factoring Trinomial - Easy Case | Step by Step

 

The general form of a quadratic trinomial is written as ax^2+bx+c where constants "a", "b" and "c"  of ax^2+bx+c are constants. The "easy" case happens when the value of "a" equals 1, that is a=1. You don't need to write the coefficient of 1 before the x^2 term because it is understood.

Thus, the general form of the "easy" case is reduced to

 

Easy Case of a Trinomial

 

diagram showing the "easy" case of trinomial as x^2+bx+c where the coefficient of the quadratic term is 1.

 

 

  • The basic strategy to factor this type of trinomial is to find two numbers (factor pair) which when multiplied, gives the constant number "c". More so, their sum (when added together) should equal to constant "b", the coefficient of the "x" term.

 

 

Diagram showing the basic strategy to factor the easy case of trinomial. Where the product of two numbers equals "c" and their sum equals "b".

 

 

Going over some examples should help you get comfortable with the steps. Let's begin!


Example 1: Factor the trinomial x^2+7x+10 as product of two binomials.

 

Obviously, this is an "easy" case because the coefficient of the squared term x is just 1. This is great! We can now focus on the steps to factor this out.

diagram showing the coefficient of the squared term is equal to 1

In this trinomial, we need to identify the other relevant constants. Observe that b=7 and c=10. Check out the correspondence below.

shows the values of: a =1, b=7, and c=10

Next, find two numbers (factor pair) that when multiplied equals the constant value of c=10, and when added equals the constant value of b=7. Because the product of two numbers must be positive, the two numbers should be both positive, or both negative.

To find the pair, you can perform several trial and error to find the correct combination. Here are some possible combinations.

 

Pair of numbers Does it equal c=10 when multiplied? Does it equal b=7 when added? Correct combination
10 and 1 Yes No wrong pair of factors
-10 and -1 Yes No wrong pair of factors
5 and 2 Yes Yes correct pair of factors
-5 and -2 Yes No wrong pair of factors

 

The only combination of numbers that can satisfy the two given requirements is the third option. The one with the green check mark. We can finally express the binomial factors of this trinomial by writing down a pair of parenthesis with an x as the leading term.

 

diagram showing where to place or position the correct combination of numbers in factored form

 

Since the correct combination of numbers are 5 and 2, our final answer should be

x^2+7x+10=(x+5)(x+2)

Check your answer using the FOIL Method to verify if the product of the two binomials give back the original trinomial.

verify that the factors are correct since (x+5) * (x+2) = x^2+2x+5x+10=x^2+7x+10

 

 


Example 2: Factor the trinomial x^2-2x-15 as product of two binomials.

 

The coefficient of the squared term is 1 so this must be the "easy" case. Now, we need to find two numbers that when multiplied result to the last constant (c=-15) , and when added give the middle constant (b=-2).

the values are a=1, b=-2 and c=-15

Since the product must be negative (value of c), the pair of numbers should have opposite signs, i.e. one number is positive while the other is negative. In addition, since the sum of two numbers is negative (value of b), the negative number must have a larger absolute value than the positive number.

Possible combinations of numbers by trial and error are listed in the following table.

 

Pair of numbers Does it equal c=-15 when multiplied? Does it equal b=-2 when added? Correct combination
15 and -1 Yes No wrong pair of factors
-15 and 1 Yes No wrong pair of factors
5 and -3 Yes No wrong pair of factors
-5 and 3 Yes Yes correct combination of factors

 

The fourth option in the table satisfies the requirements. So, the binomial factors of this trinomial is

x^2-2x-15=(x-5)(x+3)

 


Example 3: Factor the trinomial x^2+5x-24 as product of two binomials.

 

For this example, the pair of numbers must get c=-24 when multiplied, and must add up to b=5. This means that the two numbers are opposite in signs, and the positive number has larger absolute value as compared to the negative number.

 

diagram showing that the product must be -24 and the sum must be 5

 

Here is the list of possible combinations:

Pair of numbers Does it equal c=-24 when multiplied? Does it equal b=5 when added? Correct combination
-24 and 1 Yes No incorrect combination of factor pair
24 and -1 Yes No incorrect combination of factor pair
-12 and 2 Yes No incorrect combination of factor pair
12 and -2 Yes No incorrect combination of factor pair
-8 and 3 Yes No incorrect combination of factor pair
8 and -3 Yes Yes correct combination of factor pair

Since the pair of correct combination of numbers are 8 and -3 satisfies the conditions, our factored trinomial looks like this.

factoring the trinomial x^2+5x-24 as (x+8)(x-3)

 


Example 4: Factor the trinomial x62-9x+14 as product of two binomials.

 

In this case, the two numbers must have a product of c=14, and a sum of b=-9. This means that the numbers must have the same sign, either both positive or both negative. However, since the sum of the two numbers is negative, we can say that the pair should be both negative as well.

 

the product of two factors must be 14 and their sum must be equal to -9

 

By trial and error, the possible combinations are...

Pair of numbers Does it equal c=14 when multiplied? Does it equal b=-9 when added? Correct combination
14 and 1 Yes No wrong combination of two numbers
-14 and -1 Yes No wrong combination of two numbers
7 and 2 Yes No wrong combination of two numbers
-7 and -2 Yes Yes right combination of two numbers

 

Since the pair of -7 and -2 satisfies the conditions, therefore our factored trinomial looks like this.

factoring the trinomial x^2-9x+14 = (x-7)*(x-2)

 


Example 5: Factor the trinomial x62+13x+12 as product of two binomials.

Solution:

I need to find two numbers such that their product (when multiplied) equals 12, and their sum (when added) equals 13.

 

two numbers when multiplied should give 12, and when added should give 13

 

The correct combination should be 12 and 1. Why?

Because...

  • Product: (12) * (1) = 12
  • Sum: 12+1=13

The final answer is x^2+13x+12=(x+12)(x+1).

 


Example 6: Factor the trinomial x^2+8x-20 as product of two binomials.

Solution:

I need to get a product of c=-20 and a sum of b=8. Since the product is negative, the two numbers must have different signs.

Pair of numbers Does it equal c=-20 when multiplied? Does it equal b=8 when added? Correct combination
20 and -1 Yes No wrong pair of factors
-20 and 1 Yes No wrong pair of factors
5 and -4 Yes No wrong pair of factors
-5 and 4 Yes No wrong pair of factors
10 and -2 Yes Yes correct  pair of factors
-10 and 2 Yes No wrong pair of factors

Since the pair of numbers 10 and -2 satisfies the given conditions, our final answer is...

factoring the trinomial as x^2+8x-20=(x+10)(x-2)

 

 

 


Example 7: Factor the trinomial x^2-x-42 as product of two binomials.

Solution:

I need to get a product of c=-42 and a sum of b=-1. Since the product is negative, the two numbers must have different signs.

Here is the list of possible combinations of numbers.

 

Pair of numbers Does it equal c=-42 when multiplied? Does it equal b=-1 when added? Correct combination
42 and -1 Yes No wrong
-42 and 1 Yes No wrong
7 and -6 Yes No wrong
-7 and 6 Yes Yes check

 

Since the pair of numbers -7 and 6 satisfies the given conditions, our final answer is...

factoring the trinomial as x^2-x-42=(x-7)(x+6)

 

 

 


Example 8: Factor the trinomial x62-10x+21 as product of two binomials.

Solution:

I need to get a product of c=21 and a sum of b=-10. Since the product is positive, the two numbers must have the same signs, either both positive or negative.

Here is the list of possible combinations of numbers.

 

Pair of numbers Does it equal c=21 when multiplied? Does it equal b=-10 when added? Correct combination
21 and 1 Yes No incorrect
-21 and -1 Yes No incorrect
7 and 3 Yes No incorrect
-7 and -3 Yes Yes correct

 

Since the pair of numbers -7 and -3 satisfies the given conditions (product and sum) , our final answer is...

x^2-10x+27=(x-3) (x-9)

 


Example 9: Factor the trinomial -x^2+4x-4 as product of two binomials.

Solution:

The first thing that stands out is that the coefficient of x^2 term is not equal to 1. In fact, we have a=-1. First, we need to factor that -1 out of the trinomial. By doing so, the signs of each term of the trinomial will switch.

  • Factoring out -1 from -x^2+4x-4

 

factor out -1 therefore -x^2+4x-4=-1(1x^2-4x+4)

 

  • Next, focus our attention to the trinomial inside the parenthesis. At this point, it should be clear what to do since the coefficient of the x^2 is equal to 1.

 

I need to find two numbers such that their product is c=4, and their sum is b=-4.

 

two numbers that when multiplied would result to 4, and when added results to -4

 

 This is the list of possible combinations.

 

Pair of numbers Does it equal c=4 when multiplied? Does it equal b=-4 when added? Correct combination
4 and 1 Yes No wrong pair of factors
-4 and -1 Yes No wrong pair of factors
2 and 2 Yes No wrong pair of factors
-2 and -2 Yes Yes correct pair of factors

 

Since the pair of numbers -2 and -2 satisfies the given conditions, the factors would be

x^2-4x+4=(x-2)(x-2)

But don't forget the −1 that was factored out in the beginning! We need to incorporate that into our final answer which is...

-x^2+4x-4=-1(x-2)(x-2)

 


 

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