Lesson #9-1 – Adding and Subtracting Polynomials
1. Find the degree of each monomial.- 18
- 3xy3
- 6c
- –2 + 7x
- 3x5 – 2 – 2x5 + 7x
(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”
- First terms in each.
- Outside terms in each.
- Inside terms.
- Last terms.
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:
- The square of its first terms.
- + 2 x the product of its terms.
- + the square of its last terms.
FOIL algorithm.)
(x – 3)2 = (x – 3)(x – 3)
- = x2
- = 2(-3 x) = -6x
- = (-3)2
To compute a squared number use the same method.
To compute the square of 21 think of it as (20 + 1)2:
- 202 = 400
- 2(20 + 1) = 42
- 12 = 1
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:
- Factors the last term.
- Find the sum of the factors.
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
Factor1. 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)