Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview

Chapter 10

Worked example (no skipped steps)

10.1 Factor 6 𝑥 + 9.

The greatest common factor is 3:

6 𝑥 + 9 = 3(2 𝑥 + 3)

Worked example (no skipped steps)

10.2 Factor 8 𝑦 − 12.

The greatest common factor is 4:

8 𝑦 − 12 = 4(2 𝑦 − 3)

Worked example (no skipped steps)

10.3 Factor 12 𝑎 2 𝑏 + 18 𝑎𝑏.

The greatest common factor is 6 𝑎𝑏:

12 𝑎 2 𝑏 + 18 𝑎𝑏 = 6 𝑎𝑏(2 𝑎 + 3) Worked example (no skipped steps)

10.4 Factor 14 𝑚 2 − 21 𝑚.

The greatest common factor is 7 𝑚:

14 𝑚 2 − 21 𝑚 = 7 𝑚(2 𝑚 − 3)

Chapter 10

175

Worked example (no skipped steps)

10.5 Factor 𝑥 2 + 5 𝑥 + 6.

We need two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3.

So:

𝑥 2 + 5 𝑥 + 6 = ( 𝑥 + 2)( 𝑥 + 3) Worked example (no skipped steps)

10.6 Factor 𝑥 2 + 7 𝑥 + 10.

The numbers are 5 and 2 because

5 · 2 = 10 ,

5 + 2 = 7

So:

𝑥 2 + 7 𝑥 + 10 = ( 𝑥 + 5)( 𝑥 + 2) Worked example (no skipped steps)

10.7 Factor 𝑥 2 − 8 𝑥 + 15.

The numbers are −3 and −5 because

(−3)(−5) = 15 ,

−3 + (−5) = −8

So:

𝑥 2 − 8 𝑥 + 15 = ( 𝑥 − 3)( 𝑥 − 5) Worked example (no skipped steps)

10.8 Factor 𝑥 2 − 9 𝑥 + 20.

The numbers are −5 and −4.

So:

𝑥 2 − 9 𝑥 + 20 = ( 𝑥 − 5)( 𝑥 − 4) Worked example (no skipped steps)

10.9 Factor 𝑥 2 + 9 𝑥 + 20.

The numbers are 4 and 5.

So:

𝑥 2 + 9 𝑥 + 20 = ( 𝑥 + 4)( 𝑥 + 5)

176

Step-by-step solutions

Worked example (no skipped steps)

10.10 Factor 𝑥 2 − 5 𝑥 − 24.

We need two numbers that multiply to −24 and add to −5. Those numbers are −8 and 3.

So:

𝑥 2 − 5 𝑥 − 24 = ( 𝑥 − 8)( 𝑥 + 3) Worked example (no skipped steps)

10.11 Factor 𝑦 2 + 2 𝑦 − 15.

We need two numbers that multiply to −15 and add to 2. Those numbers are 5 and −3.

So:

𝑦 2 + 2 𝑦 − 15 = ( 𝑦 + 5)( 𝑦 − 3) Worked example (no skipped steps)

10.12 Factor 𝑥 2 − 16.

Recognize a difference of squares:

𝑥 2 − 16 = 𝑥 2 − 42 = ( 𝑥 − 4)( 𝑥 + 4) Worked example (no skipped steps)

10.13 Factor 𝑥 2 − 25.

Again, this is a difference of squares:

𝑥 2 − 25 = 𝑥 2 − 52 = ( 𝑥 − 5)( 𝑥 + 5) Worked example (no skipped steps)

10.14 Factor 4 𝑥 2 − 25.

Write each term as a square:

4 𝑥 2 = (2 𝑥)2 ,

25 = 52

So:

4 𝑥 2 − 25 = (2 𝑥 − 5)(2 𝑥 + 5)

Worked example (no skipped steps)

10.15 Factor 𝑎 2 − 16.

𝑎 2 − 16 = 𝑎 2 − 42 = ( 𝑎 − 4)( 𝑎 + 4)

Chapter 10

177

Worked example (no skipped steps)

10.16 Factor 9 𝑚 2 − 1.

9 𝑚 2 − 1 = (3 𝑚)2 − 12 = (3 𝑚 − 1)(3 𝑚 + 1) Worked example (no skipped steps)

10.17 Factor 3 𝑥 2 + 12 𝑥.

Take out the greatest common factor 3 𝑥:

3 𝑥 2 + 12 𝑥 = 3 𝑥( 𝑥 + 4)

Worked example (no skipped steps)

10.18 Factor 2 𝑥 2 + 10 𝑥.

Take out the greatest common factor 2 𝑥:

2 𝑥 2 + 10 𝑥 = 2 𝑥( 𝑥 + 5)

Worked example (no skipped steps)

10.19 Expand ( 𝑥 + 2)( 𝑥 + 3) to check it.

Multiply each part:

( 𝑥 + 2)( 𝑥 + 3) = 𝑥 2 + 3 𝑥 + 2 𝑥 + 6

Combine like terms:

𝑥 2 + 5 𝑥 + 6

Worked example (no skipped steps)

10.20 Expand ( 𝑥 − 4)( 𝑥 + 4) to check it.

Use distribution:

( 𝑥 − 4)( 𝑥 + 4) = 𝑥 2 + 4 𝑥 − 4 𝑥 − 16

Combine:

𝑥 2 − 16

Worked example (no skipped steps)

10.21 Decide whether 𝑥 2 + 2 𝑥 + 2 factors nicely over the integers.

Test the integer pairs for 2:

1 · 2 = 2 ,

1 + 2 = 3

(−1)(−2) = 2 ,

−1 + (−2) = −3

No pair adds to 2, so it does not factor nicely over the integers.

178

Step-by-step solutions

Worked example (no skipped steps)

10.22 Decide whether 𝑥 2 + 𝑥 − 6 factors nicely over the integers, and if so, factor it.

We need two integers that multiply to −6 and add to 1. Those integers are 3 and −2.

So:

𝑥 2 + 𝑥 − 6 = ( 𝑥 + 3)( 𝑥 − 2) Worked example (no skipped steps)

10.23 Factor 5 𝑥 2 − 45.

First take out the greatest common factor:

5 𝑥 2 − 45 = 5( 𝑥 2 − 9)

Now factor the difference of squares:

𝑥 2 − 9 = ( 𝑥 − 3)( 𝑥 + 3)

So the full factoring is:

5 𝑥 2 − 45 = 5( 𝑥 − 3)( 𝑥 + 3)

Worked example (no skipped steps)

10.24 Factor 2 𝑥 2 + 8 𝑥 + 6.

Take out the greatest common factor 2:

2 𝑥 2 + 8 𝑥 + 6 = 2( 𝑥 2 + 4 𝑥 + 3) Now factor the trinomial:

𝑥 2 + 4 𝑥 + 3 = ( 𝑥 + 1)( 𝑥 + 3) So:

2 𝑥 2 + 8 𝑥 + 6 = 2( 𝑥 + 1)( 𝑥 + 3) Chapter 11

Worked example (no skipped steps)

11.1 Explain what the point (3 , −2) means.

The first number tells the horizontal position. The second number tells the vertical position.

So (3 , −2) means: move 3 units to the right and 2 units down.

Chapter 11

179

Worked example (no skipped steps)

11.2 Find the slope through (1 , 2) and (5 , 10).

Use:

𝑦 2 − 𝑦 1

𝑚 = 𝑥 2 − 𝑥 1

Substitute:

10 − 2

8

𝑚 =

=

= 2

5 − 1

4

Answer: 𝑚 = 2.

Worked example (no skipped steps)

11.3 Find the slope through (2 , 5) and (6 , 1).

1 − 5

−4

𝑚 =

=

= −1

6 − 2

4

Answer: 𝑚 = −1.

Worked example (no skipped steps)

11.4 Find the slope through (−1 , 4) and (3 , 4).

4 − 4

0

𝑚 =

=

= 0

3 − (−1)

4

Answer: 𝑚 = 0.

Worked example (no skipped steps)

11.5 In 𝑦 = 3 𝑥 − 4, identify 𝑚 and 𝑏.

Compare with 𝑦 = 𝑚𝑥 + 𝑏. So:

𝑚 = 3 ,

𝑏 = −4

Answer: 𝑚 = 3, 𝑏 = −4.

Worked example (no skipped steps)

11.6 In 𝑦 = −2 𝑥 + 7, identify 𝑚 and 𝑏.

Compare with 𝑦 = 𝑚𝑥 + 𝑏. So:

𝑚 = −2 ,

𝑏 = 7

Answer: 𝑚 = −2, 𝑏 = 7.

180

Step-by-step solutions

Worked example (no skipped steps)

11.7 Decide whether a line with slope 0 rises, falls, or stays flat.

A slope of 0 means there is no vertical change. So the line stays flat.

Answer: stays flat.

Worked example (no skipped steps)

11.8 Explain why a vertical line has no slope.

Slope is:

change in 𝑦

𝑚 = change in 𝑥

On a vertical line, the change in 𝑥 is 0. That would mean dividing by 0, which is not allowed.

So a vertical line has undefined slope.

Worked example (no skipped steps)

11.9 Find the slope through (−2 , 4) and (3 , −6).

−6 − 4

−10

𝑚 =

=

= −2

3 − (−2)

5

Answer: 𝑚 = −2.

Worked example (no skipped steps)

11.10 Write the equation of the line with slope 2 and 𝑦-intercept −3.

Use 𝑦 = 𝑚𝑥 + 𝑏. Substitute 𝑚 = 2 and 𝑏 = −3: 𝑦 = 2 𝑥 − 3

Answer: 𝑦 = 2 𝑥 − 3.

Worked example (no skipped steps)

11.11 Write the equation of the line through (1 , 4) with slope −2.

Start with 𝑦 = 𝑚𝑥 + 𝑏 and use 𝑚 = −2: 𝑦 = −2 𝑥 + 𝑏

Now use the point (1 , 4):

4 = −2(1) + 𝑏

4 = −2 + 𝑏

Add 2 to both sides:

6 = 𝑏

So the equation is:

𝑦 = −2 𝑥 + 6

Answer: 𝑦 = −2 𝑥 + 6.

Chapter 11

181

Worked example (no skipped steps)

11.12 Decide whether the point (2 , 5) lies on the line 𝑦 = 3 𝑥 − 1.

Substitute 𝑥 = 2:

𝑦 = 3(2) − 1 = 6 − 1 = 5

The point gives 𝑦 = 5, so the point does lie on the line.

Answer: yes.

Worked example (no skipped steps)

11.13 Find the slope through (2 , 5) and (6 , 13).

13 − 5

8

𝑚 =

=

= 2

6 − 2

4

Answer: 𝑚 = 2.

Worked example (no skipped steps)

11.14 Find the slope through (−4 , −1) and (2 , 2).

2 − (−1)

3

1

𝑚 =

=

=

2 − (−4)

6

2

Answer: 𝑚 = 1 .

2

Worked example (no skipped steps)

11.15 Find the slope through (7 , −2) and (7 , 5).

5 − (−2)

7

𝑚 =

=

7 − 7

0

Division by 0 is not defined, so the slope is undefined.

Answer: the slope is undefined.

Worked example (no skipped steps)

11.16 Write the equation of the line with slope 3 and 𝑦-intercept −2 in the form 𝑦 = 𝑚𝑥 + 𝑏.

Use 𝑚 = 3 and 𝑏 = −2:

𝑦 = 3 𝑥 − 2

Answer: 𝑦 = 3 𝑥 − 2.

182

Step-by-step solutions

Worked example (no skipped steps)

11.17 Write the equation of the line with slope −1 passing through (1 , 2) in the form 𝑦 = 𝑚𝑥+ 𝑏.

Start with:

𝑦 = − 𝑥 + 𝑏

Use the point (1 , 2):

2 = −1 + 𝑏

Add 1 to both sides:

3 = 𝑏

So:

𝑦 = − 𝑥 + 3

Answer: 𝑦 = − 𝑥 + 3.

Worked example (no skipped steps)

11.18 For the line 𝑦 = 2 𝑥 − 5, find 𝑦 when 𝑥 = 4.

Substitute 𝑥 = 4:

𝑦 = 2(4) − 5 = 8 − 5 = 3

Answer: 3.

Worked example (no skipped steps)

11.19 For the line 𝑦 = 2 𝑥 − 5, find 𝑥 when 𝑦 = 9.

Set 𝑦 = 9:

9 = 2 𝑥 − 5

Add 5:

14 = 2 𝑥

Divide by 2:

𝑥 = 7

Answer: 𝑥 = 7.

Chapter 12

183

Worked example (no skipped steps)

11.20 Does the point (3 , 1) lie on the line 𝑦 = 2 𝑥 − 5?

Substitute 𝑥 = 3:

𝑦 = 2(3) − 5 = 6 − 5 = 1

The resulting 𝑦-value is 1, so the point does lie on the line.

Answer: yes.