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Unit 2A: Polynomials, Part 1

2017

pebblebrook high schoolALGBRA 2

2A.1 – 1B.6

2A.1 Classify PolynomialsVocabulary:

Polynomial

Degree of the term

Standard form of a polynomial

Degree of the polynomial

Classify using the degree of the polynomial(First Name)

Degree Name using Degree0 Constant1 Linear2 Quadratic3 Cubic4 Quadratic5 Quantic

Classify using the number of terms(Last Name)

Number of Term

Name using the number of terms

1 Monomial2 Binomial3 trinomial4 Polynomial

Example #1: Classify the polynomial. Then write in standard form

Examples of Polynomials

Classify the polynomial

Standard form

63 + x3x2

3x2+ x4

-2x + 2x3+ 5x2

-x + 2x – 3x2 – 2x5+ 4

You Try….

Classify & Write the polynomial in standard form.

1) -7x + 5x4

2) x2- 4x + 3x2+ 2x

Adding & Subtracting Polynomials…You can only add & subtract like terms (CLT)

Example #2: Add or Subtract

You Try…..

Section 2A.1 HomeworkClassify the polynomial and write in standard form.

Simplify the polynomial.

2A.2 Multiplying Polynomials

Method….Use area method (Box)Examples: Multiply

1) Multiply. Then classify the polynomial.

2) Multiply. Then classify the polynomial.

3) Multiply. Then classify the polynomial.

4) Multiply. Then classify the polynomial.

You Try….

Multiply. Then classify the polynomial.

Section 2A.2 Homework

2A.3 Word Problems: Area, Volume, & Perimeter.

Area = lw Volume = lwh Perimeter = add lengthsDon’t forget about area formulas for the other shapes…

Examples: Area & Perimeter.

Examples: Volume

1)Find the volume of the rectangular prism.

2) Find the volume of the rectangular prism.

You try…

Section 2A.3 Homework

Section 2A.4 Binomial Theorem

Multiply (x + 1)8……

Yes, there is a short cut: The Binomial Theorem.

Before we understand the pattern, let’s talk about Pascal’s Triangle.

Pascal’s Triangle is a triangular array of numbers formed by first lining the border with 1’s and then placing the sum of the two adjacent numbers within a row between and underneath the 2 original numbers.

Expand (Multiply) (a + b)4

1st: Use Pascal’s Triangle to calculate the coefficients of the 5 terms, where n = 4.

4C0 4C1 4C2

4C3 4C4

2nd: Use the Binomial Theorem to get the exponents of the terms. FIRST DECREASES; SECOND TERM INCREASES.

a4 a3b1 a2b2 ab3 b4

3rd: Make the substitutions and simply if necessary.

a4 + 4a3b1 + 6a2b2 + 4ab3 + b4

Examples: Expand

1) (x + 2)3

2) (x – 4)4

3) (v + w)9

4) (2x – 3)5

You try…

1) (x – 1)4 2) (g + h)6 3) (c – 2)5

Section 2A.4 Homework

Section 2A.5 Function OperationsA function is a relation in which each element of the domain (x-values) is paired with exactly one element in the range (y-values).

Visuals help….Wet clothes → Dryer → Dry Clothes

Input → equation → output

x → f(x)→ y

Recall…If f(x) = x + 5, thenx f(x)

You can add, subtract, & multiply functions…

Example #1: Adding, Subtracting & Multiplying FunctionsLet f(x) = 3x + 8, g(x) = 2x -12 and h(x) = -4x

1) f + g =

2) f – g =

3) g – f =

4) g * f =

5) g * h =

6) h – f =

Composite functions

In other words…

f(x) → g(x) →(g ○ f)

Example #2: Composition of a FunctionLet f(x) = x – 2 and g(x) = x2.

1) f(f(-5))

2) g(f(0))

3) f(g(-2))

4) f(g(3))

You try… Let f(x) = 3x + 5 and g(x) = x2

1) f(x) + g(x)

2) f(x) * g(x)

3) f(g(x))

Section 2A.5 Homework

Section 2A.6 Inverse Functions

In other words, the input and output switches places…

Example #1: Find the inverse from a table of points.

Example #2: Find the inverse from an equation.

Step 1: Switch x & yStep 2: solve for y.

1) y = x2 + 3

2) y = √ x+1

3) y = 2x + 6

To verify if 2 functions are inverses of one another, you must find the composition of f(g(x)) AND g(f(x)). THE BOTH MUST SIMPLIFY TO X!

Example #3: Verify the functions are inverses of one another.

1)

2)

3)

15. 15.

16.

17.

Section 2A.6 Homework