Lesson #9-1 – Adding and Subtracting Polynomials

1. Find the degree of each monomial.
  1. 18
  2. 3xy3
  3. 6c
2. Write each polynomial in standard form. Then name each by its degree.
  1. –2 + 7x
  2. 3x5 – 2 – 2x5 + 7x
3. Simplify
(6x2 + 3x + 7) + (2x2 – 6x – 4)
4. Simplify
(2x3 + 4x2 – 6) – (5x3 + 2x – 2) / Monomial – an expression that can be written as a real number, a variable, or a product of a real number and one or more variables with nonnegative powers.
Degree of a monomial – the sum of the exponents of the variables in the expression.
rt – monomial, degree 2.
-45 – monomial, degree 0.
b-5 – not (negative exponent)
-5r3 + 3r3 = (-5 + 3)r3 – monomial,
degree 3
Polynomial – an expression that is either a monomial or a sum of monomials.
Binomial – polynomial with 2 terms.
Trinomial – polynomial w/3 terms.
Degree of a polynomial – the highest degree of any of its terms after simplification.
3e5 – 2e5 + 4e7 = 7
Standard form of a polynomial – write the terms in descending order of the exponent.
Classifying Polynomials by Degree:
Linear - 4x + 3; degree 1
Quadratic – 4xy or x2 + 6; degree 2

Lesson #9-2 – Multiplying and Factoring

Simplify:
1. 4b(5b2 + b + 6)
2. –2g2(3g3 + 6g – 5)
Find the GCF of the terms:
3. 5v5+ 10v3
4. 2x4 + 10x2 – 6x.
Factor:
5. 8x2 – 12x
6. 4x3 – 8x2 + 12x / Using Area to Picture Multiplication by a Monomial:
2x(3x + 1)
x x x 1
x2 / x2 / x2 / 1
x2 / x2 / x2 / 1
x
x
= 6x2 + 2x
Finding the greatest common factor
4x3 = 2 • 2 • x • x • x
12x2 = 2 • 2 • 3 • x • x
8x = 2 • 2 • 2 • x
GCF = 2 • 2 • x
Factoring a Monomial – Find the GCF and then divide each term.
3x3 – 12x2 + 15x = 3x(x2 – 4x + 5)

Lesson #9-3 – Multiplying Binomials

1. Simplify (2y – 3)(y + 2).
2. Simplify (4x + 2)(3x – 6).
3. Find the area of the shaded region.

4. Simplify the product:
(3x2 – 2x + 3)(2x + 7) / Using Area to Picture the Multiplication of Polynomials:
(2x2 + x + 3)(3y2 + 2)
3y2 + 2
6x2y2 / 4x2
3xy2 / 2x
9y2 / 6
2x2
x
3
= 6x2y2 + 3xy2 + 4x2 + 9y2 + 2x + 6
The FOIL Algorithm”
  1. First terms in each.
  2. Outside terms in each.
  3. Inside terms.
  4. Last terms.
Works for all binomials, including those with subtraction, but you must be careful with signs.
Think of subtract as “plus a negative,” so the negative sign goes with the number right after it.
Multiply (y + 6)(y - 4).
F (first) = y  y
O (outside) = y  -4
I (inside) = 6  y
L (last) = 6  -4
Answer = y2 + (-4y) + 6y + (-24)
Simplify = y2 + 2y - 24

Lesson #9-4 – Multiplying Special Cases

Simplify:
1. (y + 11)2
2. (3w – 6)2
3. Among guinea pigs, the black fur gene (B) is dominant and the white fur gene (W) is recessive. This means that a guinea pig with at least one dominant gene (BB or BW) will have black fur. A guinea pig with two recessive genes (WW) will have white fur. You can model the probabilities with the expression (½B + ½W)2. What is the result of this product?
4. Find (p4 – 8)(p4 + 8).
5. Find 43  37. / Perfect Square Patterns:
For all number a and b,
(a + b)2 = a2 + 2ab + b2.
(a – b)2 = a2 – 2ab + b2.
The square of a binomial is:
  1. The square of its first terms.
  2. + 2 x the product of its terms.
  3. + the square of its last terms.
(OR - Just use the
FOIL algorithm.)
(x – 3)2 = (x – 3)(x – 3)
  1. = x2
  2. = 2(-3  x) = -6x
  3. = (-3)2
Equals = x2 – 6x + 9
To compute a squared number use the same method.
To compute the square of 21 think of it as (20 + 1)2:
  1. 202 = 400
  2. 2(20 + 1) = 42
  3. 12 = 1
So, 212 = 443

Lesson #9-5 – Factoring Trinomials of the Type x2 + bx + c

Factor:
1. x2 + 8x + 15
2. c2 – 9c + 20
3. x2 + 13x – 48
4. n2 – 5n - 24
5. d2 + 17dg – 60g2 / Factoring x2 + bx + c:
  1. Factors the last term.
  2. Find the sum of the factors.
b2 + 7b + 12
Factors of 12 / Sum of Factors
1 x 12 / 13
2 x 6 / 8
3 x 4 / 7
Since 3 + 4 equals 7, the factors are (b + 3)(b + 4).
If the middle and last terms are negative, then one of the factors of the factors must be negative.
a2 - 2b - 15
Factors of 15 / Sum of Factors
-1 x 15 / 14
-3 x 5 / 2
3 x –5 / -2
1 x –15 / -14
Since 3 + -5 equals –2, the factors are (a + 3)(a – 5).
If the middle term is negative and the last term is positive, then both factors must be negative.
x2 – 12x + 27
Factors of 27 / Sum of Factors
-1 x –27 / -28
-3 x –9 / -12

Lesson #9-6 – Factoring Trinomials of the Type ax2 + bx + c

Factor:
1. 2y2 + 5y + 2
2. 20x2 + 17x + 3
3. 3n2 – 7n – 6
4. 5d2 – 14d – 3
5. 18x2 + 33x – 30
6. 2v2 – 12v + 10 / Find the possible factor combina-tions to get all three coefficients.
2x2 + 9x + 9
The only factors of 2 are 1 x 2 so place these in the parenthesis.
(2x __ )(x __)
The signs are all +, so they will both be positive in the factors.
(2x + __ )(x + __)
Factors of 9: 1 x 9 and 3 x 3. Try these until you get the right combination.

Lesson #9-7 – Factoring Special Cases

Factor:
1. m2 – 6m + 9
2. The area of a square is
(16h2 + 40h + 25) in2.
Find the length of a side.
3. a2 – 16
4. 9b2 – 225
5. 5x2 – 80 /

Perfect Square Trinomials – will factor into identical binomial factors.

For every real number a and b:
a2 + 2ab + b2 = (a + b)(a + b) = (a + b)2
a2 – 2ab + b2 = (a – b)(a – b) = (a – b)2
x2 – 8x + 16 = (x – 4)(x + 4)

Lesson #9-8 – Factoring by Grouping

Factor
1. 6x3 + 3x2 – 4x – 2
2. 8t4 + 12t3+ 16t2 + 24t
3. 24h2 + 10h – 6
4. A rectangular prism has a volume of 36x3 + 51x2 + 18x.
5. 63d2 + 44d + 5
6. 11k2 + 49k + 20 / When factoring a larger polynomial you can factor by grouping.
y3 + 3y2 + 4y + 12
y2(y + 3) + 4(y + 3)
(y2 + 4)(y + 3)
Sometimes you have to factor out a monomial first.
12p4+ 10p3 – 36p2 – 30p
2p(6p3+ 5p2 – 18p1 – 15)
2p(p2(6p+ 5) – 3(6p1 – 5)
2p(p2 – 3)(6p – 5)
Copy key concepts, p 536.

Prentice Hall - Algebra 1 (2007)