Analyzing Factors of 100 and –100

8.7.3 Analyzing Factors of 100 and –100

8.7.3 • Analyzing Factors of 100 and –100

Activity

For questions 1 – 2, consider the expression $x^{2} + b x + 100$ . (Note the leading coefficient is 1.)

Create a table with all pairs of factors of +100 that would give positive values of $b$ .

For each pair, state the $b$ value they produce. (Use as many rows as needed.)

Factor 1	Factor 2	$b$ (positive)

Compare your answers:

Factor 1	Factor 2	$b$ (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 $b$ .

For each pair, state the $b$ value they produce. (Use as many rows as needed.)

Factor 1	Factor 2	$b$ (negative)

Compare your answers:

Factor 1	Factor 2	$b$ (negative)
–10	–10	–20
–20	–5	–25
–25	–4	–29
–50	–2	–52
–100	–1	–101

For questions 3 – 5, consider the expression $x^{2} + b x - 100$ . (Note the leading coefficient is 1.)

Create a table with all pairs of factors of –100 that would result in positive values of $b$ .

For each pair of factors, state the $b$ value they produce. (Use as many rows as needed.)

Factor 1	Factor 2	$b$ (positive)

Compare your answers:

Factor 1	Factor 2	$b$ (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 $b$ .

For each pair, state the $b$ value they produce. (Use as many rows as needed.)

Factor 1	Factor 2	$b$ (negative)

Compare your answers:

Factor	Factor 2	$b$ (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 $b$ .

For each pair, state the $b$ value they produce. (Use as many rows as needed.)

Factor 1	Factor 2	$b$ (zero)

Compare your answers:

Factor 1	Factor 2	$b$ (zero)
–10	10	0

For problems 6 – 9, write each expression in factored form. Note that the leading coefficient for each expression is 1.

$x^{2} - 25 x + 100$

Compare your answer:

$(x - 20) (x - 5)$

$x^{2} + 15 x - 100$

Compare your answer:

$(x + 20) (x - 5)$

$x^{2} - 15 x - 100$

Compare your answer:

$(x - 20) (x + 5)$

$x^{2} + 99 x - 100$

Compare your answer:

$(x + 100) (x - 1)$

Are you ready for more?

Extending Your Thinking

How many different integers $b$ can you find so that the expression $x^{2} + 10 x + b$ can be written in factored form?

Compare your answer:

Infinitely many. The number $b$ needs to have two factors that add up to 10. Here are some examples:

$(x + 4) (x + 6) = x^{2} + 10 x + 24$
$(x + 2) (x + 8) = x^{2} + 10 x + 16$
$(x + 11) (x - 1) = x^{2} + 10 x - 11$

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.

Access multimedia content

Self Check

Consider the expression $x^2 + bx − 48$ .

Identify all the pairs of factors of –48 that would result in positive values of $b$ .

-1 and 48,
-2 and 24,
-3 and 16,
-4 and 12,
-6 and 8
-1 and -48,
-2 and -24,
-3 and -16
-4 and -12,
-6 and -8
1 and 48,
2 and 24,
3 and 16,
4 and 12,
6 and 8
1 and -48,
2 and -24,
3 and -16,
4 and -12,
6 and -8

Additional Resources

Analyzing Factors of Constants in Quadratic Expressions

Example 1

Let’s consider the expression $x^{2} + b x + 40$ , where the leading coefficient equals 1.

Identify all the factors of 40 that would result in a positive value of $b$ .

We know that the factors of +40 that result in a positive sum are both positive numbers.

Factor 1	Factor 2	$b$ (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 $b$ .

We know that the factors of +40 that result in a negative sum are both negative numbers.

Factor 1	Factor 2	$b$ (negative)
–1	–40	–41
–2	–20	–22
–4	–10	–14
–5	–8	–13

Example 2

Now, let’s consider the expression $x^{2} + b x - 40$ , where the leading coefficient equals 1.

Identify all the factors of −40 that would result in a positive value of $b$ .

Factor 1	Factor 2	$b$ (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 $b$ .

Factor 1	Factor 2	$b$ (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 $b$ , the sum of the factors.

For positive values of $b$ , the factor with the larger absolute value is positive. The other factor is negative.

For negative values of $b$ , the factor with the larger absolute value is negative. The other factor is positive.

Try it

Analyzing Factors of Constants in Quadratic Expressions

1. Consider the expression $x^{2} + b x - 60$ . Identify all of the factors that would result in a positive value of $b$ .

Here are the factors for the expressions:

Factor 1	Factor 2	$b$ (positive)
–1	60	59
–2	30	28
–3	20	17
–4	15	11
–5	12	7
–6	10	4

This table illustrates the relationship between Factor 1 (negative integers), Factor 2 (derived as 60/|Factor 1|), and 'b' (calculated as Factor 1 + Factor 2), showing how 'b' remains positive.

2. Consider the expression $x^{2} + b x - 60$ . Identify all of the factors that would result in a negative value of $b$ .

Here are the factors for the expressions:

Factor 1	Factor 2	$b$ (negative)
1	–60	–59
2	–30	–28
3	–20	–17
4	–15	–11
5	–12	–7
6	–10	–4