Chapter 6: Quadratic Functions & Their Algebra

Topics: 1. Quadratic Function Review

2. Factoring: With Greatest Common Factor & Difference of Two Squares

3. Factoring: Trinomials

4. Complete Factoring

5. Factoring by Grouping

6. Factoring: Special Cases

LESSON 1 QUADRATIC FUNCTION REVIEW

Linear and exponential functions are used throughout mathematics and science due to their simplicity and applicability. Quadratic functions comprise another very important category of functions. You studied these extensively in Common Core Algebra I, but we will review many of their important characteristics in this unit. Exercise #1: Without the use of your calculator, evaluate each of the following quadratic functions for the specified input values. Recall that, according to the formal Order of Operations, exponent evaluation should always come first.

(a)

2

3

5

f x x

f

f

(b)

22 5

2

1

g x x

g

g

(c)

2 4

2

3

h x x x

h

h

Graphs of quadratic functions form what are known as parabolas. The simplest quadratic function, and one that you should be very familiar with, is reviewed in the next exercise.

Exercise #2: Consider the simplest of all quadratic functions 2y x .

(a) Create a table of values to plot this function over the

domain interval 3 3x . (b) Sketch a graph of this function on the grid to the right. (c) State the coordinates of the vertex of this parabola. (d) State the equation of this parabola’s axis of symmetry. (e) Over what interval is this function increasing?

x 3 2 1 0 1 2 3

2y x

y

x

QUADRATIC FUNCTIONS

Any function of the form 2f x ax bx c where the leading coefficient, a, is not zero.

All quadratic functions that have unlimited domains (domains that consist of the set of all real numbers) have a vertex (turning point) and an axis of symmetry. It is important to be able to sketch a parabola using your graphing calculator to generate a table of values.

Exercise #3: Consider the quadratic function 2 6 5f x x x .

(d) Why does this parabola open downward as

opposed to 2y x which opened upward?

(e) Between what two consecutive integers does

the larger solution to the equation 2 6 5 0x x lie? Show this point on your

graph.

Exercise #4: A sketch of the quadratic function 2 11 26y x x is shown below marked with points at its

intercepts and its vertex. Using tables or a graph on your calculator, determine the coordinates for each of the points.

The x-intercepts: A B (Zeroes)

The y-intercept: D The vertex: C

Over what interval is this function positive?

(a) Using a TABLE on your graphing calculator,

determine the vertex of this function.

(b) What is the range of this quadratic?

(c) Graph this function on the grid to the right. Use your vertex as a middle value of your graph. Graph a

few in each direction from there.

y

x

A B

C

D

x

y

LESSON 1 HOMEWORK: QUADRATIC FUNCTION REVIEW

FLUENCY

1. Without the use of your calculator, evaluate each of the following quadratic functions for the specified input values.

(a)

2 9

5

3

g x x

g

g

(b)

22 8

3

1

f x x x

f

f

(c)

2 2 6

0

2

h x x x

h

h

2. Which of the following represents the y-intercept of the quadratic function 𝑓(𝑥) = 2𝑥2 − 7𝑥 + 9? (Recall, that

the y-intercept of a graph always occurs at 𝑓(0).) (1) 7 (3) 7 (2) 2 (4) 9 3. For a particular quadratic function, the leading coefficient is negative and the function has a vertex whose

coordinates are 3,14 . Which of the following must be the range of this quadratic?

(1) | 3y y (3) | 14y y

(2) | 3y y (4) | 14y y

4. A parabola has one x-intercept of 2x and an axis of symmetry of 4x . Which of the following represents

its other x-intercept? (Hint, think of how far the given x-intercept is away from the axis.) (1) 3x (3) 6x (2) 10x (4) 8x 5. A quadratic function is shown in the table below. Which of the following statements is not true about the

function based on this table? Explain your choice. (1) The function has an x intercept of 3. (2) The function has a y-intercept of 3 . (3) The function’s leading coefficient is negative.

(4) The function has a vertex of 1, 4

x f x

1 0

0 3

1 4

2 3

3 0

4 5

5 12

6. Consider the quadratic function whose equation is 2 2 8f x x x .

(a) Sketch a graph of f on the grid provided. (b) Over what interval is f decreasing?

(c) Over what interval is 0f x ?

(d) State the range of f. APPLICATIONS 7. The number of meters above the ground, h, of a projectile fired at an initial velocity of 86 meters per second

and at an initial height of 6.2 meters is given by 24.9 86 6.2h t t t , where t represents the time, in

seconds, since the projectile was fired. If the projectile hits its peak height at 8.775t seconds, which of the following is closest to its greatest height?

(1) 265 meters (3) 422 meters (2) 384 meters (4) 578 meters 8. Physics students were modeling the height of a ball once it was dropped from the roof of a 25 story building.

The students found that the height in feet, h, of the ball above the ground as a function of the number of

seconds, t, since it was dropped was given by 2300 16h t t .

From what height was the ball dropped? To the nearest tenth of a second, determine the time at

which the ball hits the ground. Provide evidence from a table to support your answer or solve this algebraically if you recall how to.

y

x

Lesson 1 Homework Key: 1. (a) 16, 0 (b) 6, -10 (c) 6, 14

2. (4)

3. (3)

4. (2) hint: intercepts must be equidistant from the vertex

5. (3) hint: since the parabola has a minimum point at its vertex, it must point UP

6. (a) Graph below (b) 𝑥 < −1 (c) −4 < 𝑥 < 2 (d) 𝑦 ≥ −9 or [−9, ∞)

7. (2)

8. 300ft., t=4.3 seconds

LESSON 2 FACTORING WITH GCF AND DOTS

In the study of algebra there are certain skills that are called “gateway skills” because without them a student simply cannot enter into many more complex and interesting problems. Perhaps the most important gateway skill is that of factoring. The definition of factor, in two forms, is given below. Always keep in mind that when we factor (verb) a quantity, we are simply rewriting it in an different form that is completely equal to the original quantity. For example, 2 and 3 are factors of 6. It might look different, but 23 is still the number 6 just in a different form. Exercise #1: Rewrite each of the following binomials as a product of an integer with a different binomial. (a) 5 10x (b) 2 6x (c) 6 15x (d) 6 14x The above type of factoring is often referred to as “factoring out” the greatest common factor (GCF). This greatest common factor can be comprised of numbers, variables, or both. Exercise #2: Write each of the following binomials as the product of the binomial’s GCF and another binomial.

(a) 23 6x x (b)

220 5x x (c) 210 25x x (d)

230 20x

Exercise #3: Rewritten in factored form 220 36x x is equivalent to

(1) 2 10 15x x (3) 5 4 7x x

(2) 4 5 9x x (4) 9𝑥(𝑥 − 4)

FACTOR – TWO IMPORTANT MEANINGS

(1) Factor (verb) – To rewrite a quantity as an equivalent product. (2) Factor (noun) – Any individual component of a product.

Trinomials can also sometimes be factored into the product of a GCF and another trinomial. Exercise #4: Rewrite each of the following trinomials as the product of its GCF and another trinomial.

(a) 22 8 10x x (b)

210 20 5x x (c) 3 28 12 20x x x (d)

3 26 15 21x x x Another type of factoring that you should be familiar with is in the form of the difference of perfect squares. Once factored, these binomial are known as conjugate pairs: two binomials which have the same terms, but different signs. Exercise #5: Write each of the following binomials as the product of a conjugate pair.

(a) 2 9x (b)

24 x (c) 24 25x (d)

216 81x Exercise #6: Write each of the following binomials as the product of a conjugate pair.

(a) 𝑥2 −1

4 (b) 25 −

1

9𝑥2 (c)

4

81𝑥2 −

49

9 (d) 36𝑥2 − 49𝑦2

Factoring an expression until it cannot be factored anymore is known as complete factoring. Complete factoring is an important skill to master in order to solve a variety of problems. When completely factoring an expression, the first type of factoring always to consider is that of factoring out the GCF. Then, see what can be done with the remaining piece. Exercise #7: Using a combination of GCF and difference of perfect squares factoring, write each of the following in its completely factored form.

(a) 25 20x (b)

228 7x (c) 240 250x (d)

33 48x x

DIFFERENCE OF TWO SQUARES PATTERN

2 2x a x a x a

LESSON 2 HOMEWORK: FACTORING WITH GCF AND DOTS FLUENCY 1. Rewrite each of the following binomials as the product of an integer with a different binomial. (a) 10 55x (b) 24 40x (c) 6 45x (d) 18 9x 2. Rewrite each of the following binomials as the product of its GCF along with another binomial.

(a) 22 8x x (b) 6 27x (c)

230 35x x (d) 3 224 20x x

3. Rewrite each of the following binomials as the product of a conjugate pair.

(a) 2 121x (b)

264 x (c) 24 1x (d) 2 1

259

x

4. Rewrite each of the following trinomials as the product of its GCF and another trinomial.

(a) 24 12 28x x (b)

26 4 10x x (c) 3 214 35 7x x x (d)

3 220 5 15x x x 5. Completely factor each of the following binomials using a combination of GCF factoring and conjugate pairs.

(a) 26 150x (b)

236 4x (c) 228 7x (d)

327 12x x

(e) 280 125x (f)

32 200x x (g) 28 512x (h)

344 99x x

6. When completely factored, the expression 248 3x is written as

(1) 3 16 16x x (3) 3 4 4x x

(2) 3 16 16x x (4) 3 4 4x x

7. Which of the following represents the greatest common factor of the terms 2 64x y and

518xy ?

(1) 36xy (3) 52xy

(2) 2 34x y (4)

2 22x y

8. Which of the following is not a factor of 26 18x x ?

(1) 3x (3) 12 (2) 2 (4) x 9. Which of the following prime numbers is not a factor of the integer 330? (1) 11 (3) 3 (2) 7 (4) 5 APPLICATIONS 10. The area of any rectangular shape is given by the product of its width and length. If the area of a particular

rectangular garden is given by 215 35A x x and its width is given by 5x , then find an expression for the

garden’s length. Justify your response.

11. The volume of a particular rectangular box is given by the equation 350 2V x x . The height and length of

the box are shown on the diagram below. Find the width of the box in terms of x. Recall that V L W H for a rectangular box.

12. A projectile is fired from ground level such that its height, h, as a function of time, t, is given by 216 80h t t .

Written in factored form this equation is equivalent to

(1) 16 4h t t (3) 16 5h t t

(2) 8 2 7h t t (4) 8 5h t t

2x 5x

?

Lesson 2 Homework Key: 1.

2.

3.

4.

5.

6. (4) 7. (3) 8. (3) 9. (2)

10. 𝐿 = 3𝑥 − 7

11. 𝐿 = 5 − 𝑥

12. (3)

LESSON 3: FACTORING TRINOMIALS

Factoring trinomials, expressions of the form 2ax bx c , is an important skill. Trinomials can be factored if they

are the product of two binomials. The two main keys to factoring trinomials are: (1) the ability to quickly and accurately multiply binomials (FOIL) and (2) the ability to work with signed numbers. We practice both of these skills with four warm-up multiplication problems in Exercise #1.

Exercise #1: Without using your calculator, write each of the following products in simplest 2ax bx c form.

(a) 3 2 5 7x x (b) 2 3 2 5x x

(c) 5 4 2x x (d) 4 3 3 8x x

Recall from Algebra I: Case One Factoring (𝒂 = 𝟏) When factoring trinomials, we are looking for factors of the 𝑎 ⋅ 𝑐 product, which add to the 𝑏 term.

Carefully consider signs when choosing factors. Spend the time to check your choices before moving forward.

Using the factors found, re-write the trinomial as the product of two binomials.

Exercise #2: Factor each trinomial. Show all steps in the decision of factors.

(a) 2 2 35x x (b)

2 11 24x x

(c) 2 13 22x x (d)

2 5 50x x

Recall from Algebra I: Case Two Factoring (𝒂 > 1) When factoring trinomials, we are looking for factors of the 𝑎 ⋅ 𝑐 product, which add to the 𝑏 term.

Carefully consider signs when choosing factors. Spend the time to check your choices before moving forward.

Re-write the trinomial as 4 terms by splitting the middle term

Factor by grouping the polynomial into two groups of two

Rearrange by reverse distribution two re-write the trinomial as the product of two binomials.

Exercise #3: Factor each trinomial. Show all steps.

(a) 23 19 40x x (b)

22 15 18x x

(c) 215 13 2x x (d)

210 13 30x x

(e) 212 8 15x x (f)

236 35 6x x

LESSON 3 HOMEWORK: FACTORING TRINOMIALS FLUENCY 1. Multiply each of the following binomial pairs and express your answer in simplest trinomial form.

(a) 2 5 3 2x x (b) 3 8 5 1x x

(c) 8 3 7x x (d) 7 5 5 2x x

2. Which of the following is the correct factorization of the trinomial 212 23 10x x ? Hint – eliminate two of the

choices because they are “unintelligent” guesses based on the leaded term or ending term.

(1) 6 1 3 10x x (3) 4 5 3 2x x

(2) 6 2 2 5x x (4) 4 5 3 2x x

3. Written in factored form 2 16 36x x is equivalent to

(1) 3 12x x (3) 2 18x x

(2) 6 6x x (4) 9 4x x

4. Write each of the following trinomials in its factored form. Check by multiplying.

(a) 2 7 18x x (b)

2 14 24x x (c) 2 17 30x x (d)

2 5 6x x

(e) 2 5 6x x (f)

2 15 44x x (g) 2 21 20x x (h)

2 6 16x x 5. Factor each of the following with a leading coefficient other than 1. Show all steps. Check by multiplying.

(a) 25 41 8x x (b)

23 4 20x x

(c) 22 29 15x x (d)

27 39 20x x

(e) 218 25 8x x (f)

220 11 42x x

LESSON 3 HOMEWORK KEY 1. (a) 𝟔𝒙𝟐 + 𝟏𝟏𝒙 − 𝟏𝟎 (b) 𝟏𝟓𝒙𝟐 − 𝟒𝟑𝒙 + 𝟖 (c) 𝟖𝒙𝟐 + 𝟓𝟗𝒙 + 𝟐𝟏 (d) 𝟑𝟓𝒙𝟐 − 𝟏𝟏𝒙 − 𝟏𝟎 2. (4) 3. (3) 4.

5. (a) (𝟓𝒙 − 𝟏)(𝒙 − 𝟖) (b) (𝟑𝒙 + 𝟏𝟎)(𝒙 − 𝟐) (c) (𝟐𝒙 + 𝟏)(𝒙 − 𝟏𝟓) (d) (𝟕𝒙 + 𝟒)(𝒙 + 𝟓) (e) (𝟗𝒙 − 𝟖)(𝟐𝒙 − 𝟏) (f) (𝟓𝒙 + 𝟔)(𝟒𝒙 − 𝟕)

LESSON 4: COMPLETE FACTORING

Each expression that we have factored has been the product of two quantities. But, factoring can produce many more than just two factors. In Exercise #1, we first warm-up by multiplying three factors together. Exercise #1: Write each of these in their simplest form. The last two should take little time to do.

(a) 2 4 7x x (b) 5 2 5 3x x

(c) 3 5 5x x (d) 4 3 2 3 2x x x

RECALL: To completely factor an expression means to write it as a product which includes binomials that contain no greatest common factors (GCF’s).

Exercise #2: Consider the trinomial 22 4 6x x .

(a) Verify that both of the following products are

correct factorizations of this trinomial.

2 6 1x x

2 2 3x x

(b) Why are neither of these completely factored?

(c) Write each of these in completely factored form

by factoring out the GCF of each unfactored

binomial.

(d) What is true of both complete factorizations you

found in part (c)?

In practicality, it is always easiest to completely factor by ALWAYS looking for a GCF first. Once removed, the factoring then either consists of the difference of perfect squares or standard trinomial techniques. Exercise #3: Write each of the following in its completely factored form.

(a) 24 12 40x x (b)

26 24x

(c) 22 20 50x x (d)

275 3x

(e) 210 55 105x x (f)

212 57 15x x

(g) 24 5 6x x (h)

26 13 6x x

(i) 212 29 8x x

LESSON 4 HOMEWORK: COMPLETE FACTORING

FLUENCY

1. Find each of the following products in their simplest 2ax bx c form.

(a) 5 6 2x x (b) 3 2 1 2 1x x (c) 2 4 10x x x

Write each of the following expressions in their completely factored form.

Always think _______ ____________!

(2) 22 14 36x x (3)

25 70 245x x (4) 23 192x

(5) 3 26 36 96x x x (6)

328 7x x (7) 28 12 8x x

(8) 215 110 120x x (9)

3 210 26 12x x x

(10) 28 67 24x x (11)

212 20 3x x

(12) 218 39 15x x (13)

345 20x x

(14) 28 30 28x x (15)

3 290 90 20x x x

(16) 227 3x (17)

220 112 48x x

LESSON 4 HOMEWORK KEY 1. (a) 5𝑥2 − 40𝑥 + 60 (b) 12𝑥2 − 3 (c) 2𝑥3 + 28𝑥2 + 80𝑥

ALWAYS THINK GCF FIRST! 2. 2(𝑥 − 9)(𝑥 + 2)

3. 5(𝑥 + 7)(𝑥 + 7)

4. 3(𝑥 − 8)(𝑥 + 8)

5. 6𝑥(𝑥 + 8)(𝑥 − 2)

6. 7𝑥(2 − 𝑥)(2 + 𝑥)

7. 4(2𝑥 − 1)(𝑥 + 2)

8. 5(3𝑥 − 4)(𝑥 − 6)

9. 2𝑥(5𝑥 + 2)(𝑥 − 3)

10. (8𝑥 + 3)(𝑥 + 8)

11. (6𝑥 − 1)(2𝑥 − 3)

12. 3(3𝑥 + 1)(2𝑥 − 5)

13. 5𝑥(3 − 2𝑥)(3 + 2𝑥)

14. 2(4𝑥 + 7)(𝑥 + 2)

15. 10𝑥(3𝑥 − 1)(3𝑥 − 2)

16. 3(3𝑥 − 1)(3𝑥 + 1)

17. 4(5𝑥 − 2)(𝑥 + 6)

LESSON 5: FACTORING BY GROUPING

You now have essentially three types of factoring: (1) greatest common factor, (2) difference of perfect squares, and (3) trinomials. We can combine gcf factoring with the other two to completely factor quadratic expressions. Today we will introduce a new type of factoring known as factoring by grouping. This technique requires you to see structure in expressions. It is often used on polynomials of higher power. This is similar to what we do with Case II Trinomials Exercise #1: Factor a binomial common factor out of each of the following expressions. Write your final expression as the product of two binomials.

(a) 2 1 7 2 1x x x (b) 5 2 4 2x x x

(c) 5 7 7 1x x x x (d) 2 8 4 2 4x x x x

Exercise #2: Write the expression 3 4 5 3x x x as the equivalent product of binomials. Test this

equivalency with 2x .

When we factor by grouping we first extract common factors from pairs of binomials in four-term polynomials. If we are lucky, we are left with another binomial common factor. Exercise #4: Use the method of factoring by grouping to completely factor the following expressions.

(a) 3 23 2 27 18x x x (b)

3 218 9 2 1x x x

(c) 𝑥5 − 4𝑥3 + 2𝑥2 − 8 (d) 3 25 10 20 40x x x

Exercise #5: Consider the expression 2x ab ax bx .

Be careful when you use factoring by grouping. Don't force the method when it does not apply. This can lead to errors.

Exercise #6: Consider the expression 3 22 10 7 21x x x . Explain the error made in factoring it. How can you

tell that the factoring is incorrect?

3 2 2

2

2

2 10 7 21 2 5 7 3

2 7 5 3

2 7 2 8

x x x x x x

x x x

x x

(a) How can you rewrite the expression so that the

first two terms share a common factor (other

than 1)?

(b) Write this expression as an equivalent product

of binomials.

LESSON 5 HOMEWORK: FACTORING BY GROUPING

FLUENCY

1. Rewrite each of the following as the product of binomials. Be especially careful on the manipulations that involve subtraction.

(a) 5 7 5x x x (b) 4 2 3 2x x x

(c) 10 3 5 3x x x x (d) 2 7 4 4 2x x x x

(e) 4 3 2 1 2 2 1x x x x (f) 3 7 5 5 2 4x x x x

2. Max tries to simplify the expression 5 2 3 2 3 3x x x x as follows:

5 2 3 2 3 3

3 5 2 2 3

3 3 1

x x x x

x x x

x x

3. Factor each of the following quadratic expressions completely using the method of grouping:

(a) 210 6 35 21x x x (b)

212 3 20 5x x x

Explain Max’s error to him and show him the correct

simplification with all steps.

4. Factor each of the following cubic expressions completely.

(a) 3 25 2 20 8x x x (b)

3 218 27 2 3x x x

(c) 3 22 25 50x x x (d)

3 28 10 12 15x x x 5. Factor each of the following expressions. Rearrange the expressions as needed to produce binomial pairs

with common factors.

(a) 2x ac cx ax (b) xy ab ay bx

REASONING

6. Consider the expression: 3 25 9 45x x x . Enter this expression on your calculator and list all of its

zeroes (x-values where 𝑦 = 0). Then, factor it completely. Do you see the relationship between the factors and the zeroes? Explain.

LESSON 5 HOMEWORK KEY 1. (a) (𝑥 + 5)(𝑥 + 7) (b) (𝑥 − 2)(4𝑥 − 3) (c) (𝑥 − 3)(2𝑥 + 15) (d) (𝑥 + 4)(3𝑥 − 5)

(e) (2𝑥 − 1)(3𝑥 + 1) (f) (𝑥 + 5)(𝑥 + 11)

2. (𝑥 + 3)(3𝑥 + 5)Max forgot to distribute the subtraction.

3. (a) (5𝑥 + 3)(2𝑥 + 7) (b) (4𝑥 + 1)(3𝑥 − 5)

4. (a) (5𝑥 + 2)(𝑥 − 2)(𝑥 + 2) (b) (2𝑥 − 3)(3𝑥 + 1)(3𝑥 − 1)

(c) (𝑥 + 2)(𝑥 + 5)(𝑥 − 5) (d) (4𝑥 + 5)(2𝑥2 + 3)

5. (a) (𝑥 − 𝑐)(𝑥 + 𝑎) (b) (𝑦 + 𝑏)(𝑥 + 𝑎)

6. Zeros at −3, 3, 5 (𝑥 − 5)(𝑥 − 3)(𝑥 + 3) - if these were solved, it would be the same zeros.

LESSON 6: FACTORING SPECIAL CASES

There are certain types of polynomials that can be factored using short-cuts and patterns. The first type is PERFECT SQUARE TRINOMIALS. While all factorable trinomials can be factored using the techniques reviewed earlier in this chapter, there is one special type that has its own trick to it. If you can identify the pattern, it will save you some time. Be extra careful with signs!

Recognizing the pattern is the key to using this shortcut. If you recognize that the first term and last term are both perfect squares, check the middle term: it should be DOUBLE the square roots of those terms. For example: 𝑥2 + 12𝑥 + 36 You should quickly recognize that 𝑥2 and 36 are both perfect squares Check to see if the pattern works:

√𝑥2 = 𝑥 √36 = 6 2(𝑥)(6) = 12𝑥 It works!

Factor quickly, being thoughtful of the sign used. (𝑥 + 6)2

Exercise #1: Check to see if each of the following fits the ‘Perfect Square Trinomial’ pattern to factor. If it does not, factor using other methods. (a) 𝑦2 − 20𝑦 + 100 (b) 𝑎2 + 14𝑎 + 49

(c) 𝑥2 + 13𝑥 + 36 (d) 16𝑡2 − 40𝑡 + 25

(e) 4𝑥2 + 20𝑥 + 9 (f) 9𝑥2 − 24𝑥𝑦 + 16𝑦2

PERFECT SQUARE TRINOMIAL FACTORING

𝒂𝟐 + 𝟐𝒂𝒃 + 𝒃𝟐 = (𝒂 + 𝒃)𝟐

𝒂𝟐 − 𝟐𝒂𝒃 + 𝒃𝟐 = (𝒂 − 𝒃)𝟐

The second type is SUM or DIFFERENCE OF PERFECT CUBES. For these types, this is the only form of factoring we will consider in this chapter. Be extra careful with signs!

First, let’s recall some commonly seen cubes: (1)3 = ____ (2)3 = ____ (3)3 = ____ (4)3 = ____ (5)3 = ____ (6)3 = ____ (7)3 = ____ (8)3 = ____ (9)3 = ____ (10)3 = ____ For example:

(8𝑥3 + 27) You should quickly recognize that 8𝑥3 and 27 are both perfect cubes and this is SUM

Identify the 𝑎 and 𝑏 terms

8𝑥3 = (2𝑥)3 27 = (3)3

Apply the pattern, ensure proper signs are used throughout.

𝑎3 + 𝑏3 = (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏2)

8𝑥3 + 27 =

Exercise #2: Factor using sum or difference of perfect cubes. (a) 𝑡3 + 1 (b) 64𝑥3 + 𝑦3

(c) 27𝑚3 − 8 (d) 27𝑥3 − 64𝑦3

(e) 𝑥3 + 8𝑦3 (f) 125𝑔3 − ℎ3

SUM OR DIFFERENCE OF PERFECT CUBES

𝒂𝟑 + 𝒃𝟑 = (𝒂 + 𝒃)(𝒂𝟐 − 𝒂𝒃 + 𝒃𝟐)

𝒂𝟑 − 𝒃𝟑 = (𝒂 − 𝒃)(𝒂𝟐 + 𝒂𝒃 + 𝒃𝟐)

same signs

opposite signs

always positive

LESSON 6 HOMEWORK: FACTORING SPECIAL CASES FLUENCY

1. Factor each of the following using Perfect Square Trinomial factoring.

(a) 𝑎2 + 4𝑎 + 4 (b) 𝑝2 + 2𝑝 + 1

(c) 𝑘2 + 22𝑘 + 121 (d) 25𝑘2 − 20𝑘 + 4

(e) 4𝑥2 + 12𝑥 + 9 (f) 100𝑎2 − 140𝑎𝑏 + 49𝑏2

(g) 49𝑥2 + 28𝑥𝑦 + 4𝑦2 (h) 𝑦2 − 8𝑦 + 16

2. Factor each of the following using Sum or Difference of Cubes. Write the pattern equation each time

before plugging in.

(a) 𝑎3 − 1 (b) 𝑥3 − 8

(c) 𝑝3+𝑞3 (d) 𝑘3 − 𝑓3

(e) 𝑦3 − 8𝑥3 (f) 27𝑥3 − 1

(g) 125𝑡3 + 8𝑟3 (h) 216𝑧3 − 27𝑤3

LESSON 6 HOMEWORK: FACTORING SPECIAL CASES 1.

a) (𝑎 + 2)2

b) (𝑝 + 1)2

c) (𝑘 + 11)2

d) (5𝑘 + 2)2

e) (2𝑥 + 3)2

f) (10𝑎 + 7𝑏)2

g) (7𝑥 + 2𝑦)2

h) (𝑦 + 4)2

2. a) (𝑎 − 1)(𝑎2 + 𝑎 + 1)

b) (𝑥 − 2)(𝑥2 + 2𝑥 + 4)

c) (𝑝 + 𝑞)(𝑝2 − 𝑝𝑞 + 𝑞2)

d) (𝑘 − 𝑓)(𝑘2 + 𝑘𝑓 + 𝑓2)

e) (𝑦 − 2𝑥)(𝑦2 + 2𝑥𝑦 + 4𝑥2)

f) (3𝑥 − 1)(9𝑥2 + 3𝑥 + 1)

g) (5𝑡 + 2𝑟)(25𝑡2 − 10𝑡𝑟 + 4𝑟2)

h) (6𝑧 − 3𝑤)(36𝑧2 + 18𝑧𝑤 + 9𝑤2)