Chapter 10 Notes

Day 15 10.1– Adding and Subtracting Polynomials 10.2 – Multiplying Polynomials 10.1 Classifying Polynomials Polynomial: Usually written in ________________________________ Degree: Degree of a Polynomial: Leading Coefficient: Names for polynomials with different degrees are _________________, ___________________, _____________________, ___________________, and _____________________. Names for polynomials with different numbers of terms are _________________, __________________, and __________________.

Classify the following Polynomials: Polynomial Degree Type (by degree)

Leading Type (by terms) Coefficient

4 -3x + 2 2x 4x3 – 2x2 + 3x +2 3x2 + 1 x4 1 – 5x + 7x3 8x3 + 7 – x2 3x – 15x2 21 Adding & Subtracting To simplify, we must add or subtract ___________________________________ (have the same degree).

1. Find the sum of (5x3 – x + 2x2 + 7) + (x + x2 + 6) Vertically:

Horizontally:

Recall: The Distributive Property! 2. Find the difference of (3x2 – 5x + 3) – (2x2 – x – 4) Vertically:

Horizontally:

3. (2a2 – 4a + 3) + (6a2 + 4a – 3)

4. (5x2 + 2x – 1) + 8x2

5. (-3 + 2n2 + 5n5) – (4 – n3 + 2n2 + n5)

6. (6x – 3x2 + 1) – (9x – 4 – 3x2)

7. (6x – 5) – (8x + 15) + (3x – 4)

8. -10(u + 1) + 8(u – 1) – 3(u + 6)

10.2 Multiplying Polynomials Recall (again): The Distributive Property! 9. 3x(2x2 – 5x + 3) =

10. (2x – 3)(x + 7) =

Geometric Example:

w= h= Square Area (add up each square) =

Area of the entire rectangular region = F= O= I= L= 11. (3x – 4)(2x + 1) =

12. (x – 3)(2x + 2) =

13. (5a – b)(a + b) =

14. (2m – 1)(2m – 5) =

15. (2y – 4z)(2y + z) =

16. (x + 4)(3x + 2) =

Binomial-Trinomial & 3-Binomal Multiplication Caution!! 17. (x – 2)(5 + 3x – x2) = Vertically:

Horizontally:

18. (-x2 + 2x + 4)(x – 3) =

19. (x + 1)(5x3 –x2 + x + 4) =

20. (x – 1)(x3 + 2x2 + 2) =

21. (a2 - ab + b2)(a + b) =

22. (x + 9)(x – 2)(x – 7) =

23. (2x – 6)(x + 7)(x + 6) =

Day 16 10.3 – Special Products 10.4 – Solving in Factored Form Review: Simplify. 1. (2x2 – 3x – 1) – (5x2 + 2x – 3) =

2. 3x2(x3 – 6x2 + x – 5) =

3. (x – 6)(3x2 – 5) =

4. (x2 – 8x – 3)(x + 2) =

10.3 Special Products Recall: F. O. I. L. (see #3 above) Today: Special Binomial-Binomial Multiplication Patterns (a.k.a. You can’t fool me…I know what’s going to happen) Complete the examples and determine the patterns. Example: (4x + 3)(4x + 3) = Pattern: (a + b)2 = (a + b)(a + b) =

Example: (2x – 5)(2x – 5) =

Pattern: (a – b)2 = (a – b)(a – b) =

Example: (x – 3)(x + 3) =

Pattern: (a + b)(a – b) =

Geometric Example of (a + b)(a + b) = w= h= Square Area (add up each square) =

Area of the entire rectangular region = Yay, practice! 5. (x – 2)(x + 2) =

6. (4x + 7)(4x – 7) =

7. (2a – 1)2 =

8. (x + y)2 =

9. (3m + 5)2 =

10. (3 – 10z)2 =

10.4: Solving in Factored Form Recall: We can solve Quadratics by * * Today: Factored form! Zero Product Property: If ab = 0, then _________________ or _________________.

Solve (x – 2)(x + 3) = 0.  When the equation equals zero, that means “solve!” Why?

11. (2x – 3)(x + 5) = 0

12. 5x(2x + 3) = 0

13. 2(x – 5)2 = 0

14. Cubic: 3x(4x + 2)(9x + 18) = 0

Recall: Finding the Vertex and Axis of Symmetry Today: x- coordinate of vertex is ____________________________________________. Find the roots, vertex, and axis of symmetry of y = (x – 2)(x – 6). Graph (label two additional points on your graph).

Find the roots, vertex, and axis of symmetry. Graph (label two additional points on your graph). 15. y = 4(x + 1)(x – 1) 16. y = - (x + 3)(x + 5)

Day 17 10.5 – Factoring the Greatest Common Factor and x2 + bx + c 10.7 – Factoring Special Products Review. 1. (13b + q)2

2. (10pq + 5r)(10pq – 5r)

3. 5x(4x – 1)(8x -7) = 0

4. Find the vertex of y = (x + 5)(x – 2)

10.5: Factoring the GCF Recall: Zero – Product & Distributive Properties Today: Factoring in order to solve using the Zero – Product Property Think about GCF factoring as ‘reverse distribution’ Before: y = 6(x + 2) = 6x + 12

Now: y = 6x + 12 = 6(x + 2) Solve: 6(x + 2) = 0 x+2=0 x = -2

Before: y = 5b(2b – 3) =

Now: y = 10b2 – 15b = Solve:

What is the GCF of all coefficients? What is the lowest exponent of all of the terms? Factor. 1. 16x2y3 – 12x3y2

2. 7r5s2 + 49r2s

3. 16wv4 + 12w3v2

4. 2d2e2 – 8d6e6

10.5: Factoring x2 + bx + c All of the trinomials we will factor today have an x2 coefficient of ___________! Next class we will factor trinomials with other values for a. Ex 1: Factor and solve x2 + x – 6 = 0 Step 1: Find a*c Find b

a*c = b=

Step 2: Find two numbers that have a product of the a*c value and a sum of the b value Step 3: Write the trinomial as the product of two binomials Step 4: Solve, if necessary.

Ex 2: Factor and solve x2 + 9x + 18 = 0 Step 1: Find a*c Find b

a*c = b=

Step 2: Find two numbers that have a product of the a*c value and a sum of the b value Step 3: Write the trinomial as the product of two binomials Step 4: Solve, if necessary.

Ex 3: Factor and solve x2 + 5x – 36 = 0

Factor. 5. x2 + 10x + 21

6. x2 + 8x + 15

7. x2 – 9x + 20

8. x2 – 8x – 9

9. x2 + 3x – 18

10. x2 + 6x – 5

How can you tell whether or not a trinomial can be factored?

Factor and solve. If not factorable, provide evidence. 11. x2 + 11x = -18 12. x2 – 5x = 24

13. x2 + 3x = 14

14. x2 – 5x = 32

10.7: Factoring Special Products Recall: Special Binomial-Binomial Multiplication Patterns Ex 4: Factor x2 – 36

Pattern:

Ex 5: Factor x2 + 6x + 9

Pattern:

Ex 6: Factor x2 – 12x + 36

Pattern:

Note: Factor. 15. 4x2 – 25

16. 16y2 – 24y + 9

17. 2x2 + 8x + 8

18. x2 + 9

19. 12 – 27x2

20. 9y2 + 60y + 100

21. 2x2 – 12x + 18

22. 4n2 – 16m2

23. If you kick a ball with an initial velocity of 32 ft/sec, will the ball reach a height of 16 feet? If it does, how long will it take to reach that height? Formula: Model: Factor and solve:

Day 18 11.6 – Factoring ax2 + bx + c 11.8 – Factoring with four terms Review: Factor and solve. If not factorable, provide evidence. 1. x2 – 10x + 24 = 0

2. x2 + 10x = - 9

3. 3x2 + 18x = 120

4. 12x2 + 27x = 0

5. 36x2 – 25 = 0

6. 4x2 – 40x = - 100

Recall: Steps to factor x2 + bx + c Step 1: Step 2: Step 3: Step 4: Today: We will factor trinomials of the form ax2 + bx + c where the coefficient of x2 is a number ________________________________________________________________. Today’s style of factoring is called the British method.

Ex 1: Factor and solve 6x2 – x – 2 = 0 Step 1: Find a*c a*c = Find b b= Step 2: Find two numbers that have a product of the a*c value and a sum of the b value Step 3: Write the trinomial as a polynomial with four terms, splitting the bx term into two terms using the values from Step 3 Step 4: Group the first two terms and the last two terms, inserting an addition sign in the middle. Step 5: Factor the GCF from the first group and factor the GCF from the second group. Step 6: Checkpoint! Factor the matching binomials. Group the outside terms. Step 7: Solve, if necessary. Ex 2: Factor and solve 8x2 – 2x – 3 = 0 Step 1: Find a*c a*c = Find b b= Step 2: Find two numbers that have a product of the a*c value and a sum of the b value Step 3: Write the trinomial as a polynomial with four terms, splitting the bx term into two terms using the values from Step 3 Step 4: Group the first two terms and the last two terms, inserting an addition sign in the middle. Step 5: Factor the GCF from the first group and factor the GCF from the second group. Step 7: Checkpoint! Factor the matching binomials. Group the outside terms. Step 8: Solve, if necessary.

Practice! Factor completely. Solve, if necessary. 1. 3x2 + 13x + 14

2. 5x2 + 27x – 18

3. 3x2 + 5x + 2

4. 2x2 + 21x – 11

5. 8x2 – 14x – 15

6. 6x2 + 9x – 27

7. 7x2 – 4x – 3

8. 6x2 – 5x – 21

9. 10x2 + 25x – 90

10. 8x2 + 10x – 11 = 12x + 10

Factoring by grouping:

13. x3 + 4x2 + 6x + 24

14. x3 + 2x2 – 36x – 72

Factor completely. Completely means: 15. 4x2 – 36

16. 75x4 – 3x2

17. 5x3 – 25x2 – 30x

18. 2x3 + 3x2 – 50x – 75

19. If the diving board at the Gunn pool is 2 feet above the water and you jump off the platform with an initial velocity of 4 feet per second, how long will it take you to enter the water? Model: Factor to solve:

Day 19 Chapter 10 Review 1. (- z3 + 3z) + (- z2 – 4z – 6)

2. (5x2 + 7x – 4) – (4x2 – 2x)

3. 5b2(3b3 – 2b2 + 1)

4. (3z + 4)(5z – 8)

5. (3 + 2s – s2)(s – 1)

6. (5p – 6q)2

7. (10x – 5y)(10x + 5y)

8. 3(x + 2)2 = 0

9. 2(3x – 1)(2x + 5) = 0

10. Find the x – intercepts, vertex, and axis of symmetry of the function. Graph (find two additional points for your graph). y = (x – 3)(x + 1)

Solve by factoring. 11. x2 – 2x = 24

12. 3x2 – 5 = - 14x

Factor completely. 13. x4 + 4x3 – 45x2

14. 8t3 – 3t2 + 16t – 6

Day 20 Quest today on Chapter 10 Review. 1. (3a + 2a4 – 5) – (a3 + 2a4 + 5a)

2. 4x2 – 5x = 6

3. 121 – x2

4. 64y2 + 48y + 9

5. x4 + 4x3 – 45x2 = 0

6. x3 + 2x2 – 4x - 8