Name



Algebra Chapter 8 Notes

Polynomials and Factoring

Lesson 8.1 Add and Subtract Polynomials

Monomial: __________________________________________________________________________

____________________________________________________________________________________

Degree of a monomial: ________________________________________________________________

Polynomial: _________________________________________________________________________

____________________________________________________________________________________

Degree of a polynomial: _______________________________________________________________

Leading coefficient: __________________________________________________________________

____________________________________________________________________________________

Binomial: ___________________________________________________________________________

Trinomial: __________________________________________________________________________

Example 1: Write the polynomials so that the exponents decrease from left to right. Identify the degree and the leading coefficient of the polynomial.

9 – 2x2 16 + 3y3 + 2y 6z3 + 7z2 – 3z5 12x3 – 15x + 13x5

Example 2: Tell whether the expression is a polynomial. If it is a polynomial, find its degree and classify it by the number of its terms. Otherwise, tell why it is not a polynomial.

5xy2 3a-5 x4 + 3x3 – x [pic] 6a2c + 5ac5

Example 3: Find the sum.

(3x4 – 2x3 + 5x) + (7x2 + 9x3 – 2x) (7x2 – 3x + 6) + (9x2 + 6x – 11)

Example 4: Find the difference.

(3x2 – 9x) – (2x2 – 5x + 6) (11x2 + 6x – 1) – (2x2 – 7x + 5)

Example 5: During the period 1999-2005, the number of hours an individual person watched broadcast television B and cable and satellite television C can be modeled by B = 2.8t2 – 35t + 879 and C = -5t2 + 80t + 712 where t is the number of years since 1999. About how many hours did people watch television in 2002?

Lesson 8.2 Multiply Polynomials

Example 1: Find the products.

5x 4(2x3 – 3x2 + x – 6) 3x2(7x2 – 2x + 3) 4x(3x3 – 2x2 – 8x + 9)

Example 2: Find the product.

(5m2 – 2m + 3)(2m + 7)

Example 3: Find the product.

(9x2 – x + 6)(5x – 2)

Example 4: Multiply binomials using FOIL pattern.

(2x – 1)(7x + 6) (2n + 7)(3n + 4) (8s – 7)(9s – 7)

Example 5: The dimensions of a rectangle are 3x – 1 and x + 5. Which expression represents the area of the rectangle?

Lesson 8.3 Find Special Products of Polynomials

Square of a Binomial Pattern

Algebra Example

(a + b)2 = a2 + 2ab + b2 (x + 3)2 =

(a – b)2 = a2 – 2ab + b2 (3x – 2)2 =

Example 1: Find the product using the square of a binomial patterns.

(7x + 2)2 (6x – 5y)2

Sum and Difference Pattern

Algebra Example

(a + b)(a – b) = a2 – b2 (x + 5)(x – 5) =

Example 2: Find the products using the sum and difference pattern.

(m + 9)(m – 9) (4n – 3)(4n + 3)

Example 3: Use special products to find the products.

37. • 43 552 31 • 49

Example 4: In pea plants, the gene G is for green pods, and the gene y is for yellow pods. Any gene combination with a G results in a green pod. Suppose two pea plants have the same gene combination Gy. The Punnett square shows the possible gene combinations of an offspring pea plant and the resulting pod color.

a. What percent of possible gene combinations of the offspring plant result in a yellow pod?

b. Show how you could use a polynomial to model the possible gene combinations of the offspring.

[pic]

Lesson 8.4 Solve Polynomial Equations in Factored Form

Zero-Product Property: ________________________________________________________________

Roots: ______________________________________________________________________________

________________________________________________________________________________________________________________________________________________________________________

Vertical motion model: ________________________________________________________________

________________________________________________________________________________________________________________________________________________________________________

Example 1: Use the zero-product property to solve the equations.

(x – 3)(x + 6) = 0 (m – 7)(m – 9) = 0 (5n + 10)(4n + 12) = 0

Example 2: Factor out the greatest common monomial factor.

8x +12y 14y2 + 21y 4x4 – 24x3

Example 3: Solve the equation by factoring first.

6x2 + 12x = 0 4k2 – 8k = 0 9y2 = 21y

Example 4: A child jumping rope leaves the ground at an initial vertical velocity of 8 feet per second. After how many seconds does the child land on the ground?

In Example 4, suppose the initial velocity is 10 feet per second. After how many seconds will the child land on the ground?

Lesson 8.5 Factor x2 + bx + c

Factoring x2 + bx + c:

x2 + bx + c = (x + p)(x + q) provided ______________________________________________________

x2 + 5x + 6 = _________________________________________________________________________

Example 1: Factor.

y2 + 6y + 5 x2 + 10x + 16 x2 + 10x + 24

Example 2: Factor.

z2 – 7z + 12 w2 – 10w + 9 x2 – 8x + 12

Example 3: Factor.

k2 + 6k – 7 y2 + 2y – 63 z2 – 5z – 36

Example 4: Solve the equations.

h2 – 4h = 21 x2 + 30 = 11x

Lesson 8.6 Factor ax2 + bx + c

5 step method for factoring ax2 + bx + c:

1. ______________________________________________________________________________

2. ______________________________________________________________________________

3. ______________________________________________________________________________

4. ______________________________________________________________________________

5. ______________________________________________________________________________

Example 1: Factor.

5n2 – 12n + 7 7a2 – 50a + 7

Example 2: Factor.

3m2 – 5m – 22 4b2 – 8b – 5

Example 3: Factor.

–2x2 + 9x – 9 –3r2 – 7r – 4

Example 5: A rectangle’s length is 5 feet more than 4 times its width. The area is 6 square feet. What is the width?

Lesson 8.7 Factor special products.

Perfect square trinomials: ______________________________________________________________

________________________________________________________________________________________________________________________________________________________________________

Difference of Two Squares Pattern:

a2 – b2 = (a + b)(a – b) x2 – 121 = _____________________________________________

Example 1: Factor the polynomials.

r2 – 81 9s2 – 4t2 80 – 125q2

Perfect Square Trinomial Pattern:

a2 + 2ab + b2 = (a + b)2 x2 + 6x + 9 = __________________________________________

a2 – 2ab + b2 = (a – b)2 x2 – 10x + 25 = ________________________________________

Example 2: Factor the polynomial.

x 2 + 14x +49 144y2 – 120y + 25

150z2 – 60z + 6 -2x2 – 16x – 32

Example 3: Solve the equation.

q2 – 100 = 0 r2 – 10r + 25 = 0 16m2 – 81 = 0

Lesson 8.8 Factor Polynomials Completely

Factor by grouping: ___________________________________________________________________

________________________________________________________________________________________________________________________________________________________________________

Prime polynomial: ____________________________________________________________________

____________________________________________________________________________________

Factored completely: __________________________________________________________________

________________________________________________________________________________________________________________________________________________________________________

Example 1: Factor the expression by finding a common binomial.

5x2(x – 2) – 3(x – 2) 7y(5 – y) + 3(y – 5) 11x(x – 8) + 3(x – 8)

Example 2: Factor the polynomial by grouping.

m3 + 7m2 – 2m – 14 n3 + 30 + 6n2 + 5n 10x3 + 21y – 35x2 – 6xy

Example 3: Factor the polynomial completely.

x2 – 4x – 3 3x3 – 21x2 – 54x 8d3 + 24d

Example 4: Solve the polynomial equation.

2c3 + 8c2 – 42c = 0 4x3 + 48x2 + 144x = 0 7x3 + 14x2 = 105x

Example 5: A box of crackers has a volume of 12 cubic inches. The box has a height of x inches, width of (x – 1) inches, and a length of (x + 4) inches. Find the dimensions of the box.

-----------------------

Algebra 1

Assessment Book

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download