Activity
For questions 1 - 2, consider the expression . (Note the leading coefficient is 1.)
Create a table with all pairs of factors of +100 that would give positive values of .
For each pair, state the value they produce. (Use as many rows as needed.)
Factor 1 | Factor 2 | (positive) |
Compare your answers:
Factor 1 | Factor 2 | (positive) |
10 |
10 |
20 |
20 |
5 |
25 |
25 |
4 |
29 |
50 |
2 |
52 |
100 |
1 |
101 |
Create a table with all pairs of factors of +100 that would give negative values of .
For each pair, state the value they produce. (Use as many rows as needed.)
Factor 1 | Factor 2 | (negative) |
Compare your answers:
Factor 1 | Factor 2 | (negative) |
–10 |
–10 |
–20 |
–20 |
–5 |
–25 |
–25 |
–4 |
–29 |
–50 |
–2 |
–52 |
–100 |
–1 |
–101 |
For questions 3 – 5, consider the expression . (Note the leading coefficient is 1.)
Create a table with all pairs of factors of –100 that would result in positive values of .
For each pair of factors, state the value they produce. (Use as many rows as needed.)
Factor 1 | Factor 2 | (positive) |
Compare your answers:
Factor 1 | Factor 2 | (positive) |
25 |
–4 |
21 |
20 |
–5 |
15 |
50 |
–2 |
48 |
100 |
–1 |
99 |
Create a table with all pairs of factors of –100 that would give negative values of .
For each pair, state the value they produce. (Use as many rows as needed.)
Factor 1 | Factor 2 | (negative) |
Compare your answers:
Factor | Factor 2 | (negative) |
–25 |
4 |
–21 |
–20 |
5 |
–15 |
–50 |
2 |
–48 |
–100 |
1 |
–99 |
Create a table with all pairs of factors of –100 that would result in a zero value of .
For each pair, state the value they produce. (Use as many rows as needed.)
Factor 1 | Factor 2 | (zero) |
Compare your answers:
Factor 1 | Factor 2 | (zero) |
–10 |
10 |
0 |
For problems 6 – 9, write each expression in factored form. Note that the leading coefficient for each expression is 1.
Enter the expression in factored form.
Compare your answer:
Enter the expression in factored form.
Compare your answer:
Enter the expression in factored form.
Compare your answer:
Enter the expression in factored form.
Compare your answer:
Are you ready for more?
Extending Your Thinking
How many different integers can you find so that the expression can be written in factored form?
Enter the number of different integers .
Compare your answer:
Infinitely many. The number needs to have two factors that add up to 10. Here are some examples:
There are infinitely many examples because there are infinitely many pairs of integers whose sum is 10.
Video: Analyzing factors of 100 and –100
Watch the following video to learn more about analyzing the factors of 100 and –100 and applying them in factoring quadratic expressions.
Self Check
Additional Resources
Analyzing Factors of Constants in Quadratic Expressions
Example 1
Let’s consider the expression , where the leading coefficient equals 1.
Identify all the factors of 40 that would result in a positive value of .
We know that the factors of +40 that result in a positive sum are both positive numbers.
Factor 1 | Factor 2 | (positive) |
1 |
40 |
41 |
2 |
20 |
22 |
4 |
10 |
14 |
5 |
8 |
13 |
Now let’s identify all the factors of 40 that would result in a negative value of .
We know that the factors of +40 that result in a negative sum are both negative numbers.
Factor 1 | Factor 2 | (negative) |
–1 |
–40 |
–41 |
–2 |
–20 |
–22 |
–4 |
–10 |
–14 |
–5 |
–8 |
–13 |
Example 2
Now, let’s consider the expression , where the leading coefficient equals 1.
Identify all the factors of −40 that would result in a positive value of .
Factor 1 | Factor 2 | (positive) |
–1 |
40 |
39 |
–2 |
20 |
18 |
–4 |
10 |
6 |
–5 |
8 |
3 |
Identify all the factors of –40 that would result in a negative value of .
Factor 1 | Factor 2 | (negative) |
1 |
–40 |
–39 |
2 |
–20 |
–18 |
4 |
–10 |
–6 |
5 |
–8 |
–3 |
Notice the similarities between the two tables above. They both contain all of the factors of –40, where one factor is positive and the other is negative.
Notice the difference depends on the value of , the sum of the factors.
- For positive values of , the factor with the larger absolute value is positive. The other factor is negative.
- For negative values of , the factor with the larger absolute value is negative. The other factor is positive.
Try it
Try It: Analyzing Factors of Constants in Quadratic Expressions
1. Consider the expression . Identify all of the factors that would result in a positive value of .
Here are the factors for the expressions:
Factor 1 |
Factor 2 |
(positive) |
–1 |
60 |
59 |
–2 |
30 |
28 |
–3 |
20 |
17 |
–4 |
15 |
11 |
–5 |
12 |
7 |
–6 |
10 |
4 |
2. Consider the expression . Identify all of the factors that would result in a negative value of .
Here are the factors for the expressions:
Factor 1 | Factor 2 | (negative) |
1 |
–60 |
–59 |
2 |
–30 |
–28 |
3 |
–20 |
–17 |
4 |
–15 |
–11 |
5 |
–12 |
–7 |
6 |
–10 |
–4 |