Algebra II Chapter 5 Notes

Algebra II Chapter 5 Notes

Day 1: 5.1 Use Properties of Exponents

1. Product of powers: When the bases are the_____– add the exponents

Ex.1:

2. Power of a power: When have one base with ____ powers - ______the exponents.

Ex.2:

3. Power of a product: When two bases with _____ exponent - ______the bases and raise

to power of the exponent.

Ex.3:

4. Negative exponent: When have a base with a ______exponent- _____ the base and

raise to positive power. DO NOT make the answer negative.

Ex.4:

5. Zero exponent: Anything raised to the _____ power is 1.

Ex.5:

6. Quotient of powers: When the bases are the same in a division problem -______the

exponents.

Ex.6:

Try these:

1. 2. 3. =

4. 5. 6.

Scientific Notation: Use properties of exponents

Ex. 1:

Ex. 2:

Day 26:5.2 Evaluate and graph polynomial functions

Polynomial function:

Where: and the exponents are all _____ numbers, and the coefficients are all _____ numbers.

Leading coefficient: The coefficient of the term with the highest exponent.

Degree: The degree is the value of the highest exponent.

Common Polynomial Functions
Degree / Type / Standard Form / Example
0 / Constant / /
1 / Linear / /
2 / Quadratic / /
3 / Cubic / /
4 / Quartic / /

Decide whether the function is a polynomial function. If so write in standard form, state its degree, type, and leading coefficient.

1. 2.

3. 4.

Evaluate by direct substitution: Plug in value given for x into function wherever x is.

Use direct substitution to evaluate when x=3.

Try: #1: #2:

Synthetic substitution: How to evaluate a function more easily than direct substitution

Use synthetic substitution to evaluate when x=3.

Step 1: Write coefficients in descending order using zeros for terms missing.

3 2 -5 0 -4 8

Step 2: Bring down first coefficient. Multiply that by x-value. Write the product under the second coefficient. Repeat until done.

Step 3: Final sum is value of f(x) when x = 3.

Try these: #1:

#2:

End Behavior: Behavior of the graph of a function as x approaches positive infinity or negative infinity .It is determined by the function’s ______and the sign of the ______coefficient.

End Behavior of Graphs of Polynomial Functions

Graphing polynomial functions: Make a table of values and plot the points. Connect the points into a smooth curve and check end behaviors.

Graph

x / -3 / -2 / -1 / 0 / 1 / 2 / 3
y / / / /

Day 27:5-3Add,Subtract,and multiply polynomials

Add polynomials: collect like terms; put answer in descending order.

Ex: Add and =

(horizontal)

Or (vertical)

Ex: Subtract from

Horizontally: () - ()=

Always distribute the negativefirst then add.

Vertically:

- ()

Ex:Multiply and

Box method keeps you better organized.

Combine like terms in diagonals.

Ex: Multiply 3 binomials: x-5,x+1, and x+3.

Multiply first two by foiling. Then multiply that product by third binomial using box.

(x – 5)(x+1)= ( ) (x + 3) =

Special product patterns:

1. Sum and difference: = []

2. Square of a binomial:

3. Cube of a binomial:

5.5 Polynomial long division

Step 1: Put polynomial in standard form and include “0” coefficient for missing terms

Step 2: Divide first term of polynomial by first term of divisor each time until done

Step 3: Remember the order: DMSB

Step 4: Express the remainder as a fraction with the denominator being the divisor

Ex: Divide by

Try: Divide by

Day 29:5-4 Factor and solve polynomial equations

Find a common monomial factor:

1. Factor: 2. 3.

Factor: Sum of two cubes: The trinomial will never factor.

1. 2. =

Factor: Difference of two cubes:

1. = 2. = 3.

Factor by grouping:

1. 2. =

To solve --- set each factor = 0 and solve for the variable.

Find the real number solutions of:

1. 2.

Day 30:5-5 Apply the remainder and factor theorems

Remainder Theorem: If a polynomial is divided by a factor(x-k) then the remainder(r) is = f(k).

Use synthetic division: Divide by

k = -3

-3 2 1 -8 5

Notice that according to the remainder theorem.

Try: divide by

Factor theorem: If f(k) = 0 then (x-k) is a factor of f(x). When you use synthetic division you will have ___ remainder if (x-k) is a factor of f(x).

Ex#1 Given one ______of a polynomial, find the other factors.

Hint: Use synthetic division to factor the polynomial until you can factor without synthetic division.

Factor completely given that is a factor.

-2 3 -4 -28 -16

Fill in variables in descending order starting with one degree lower .

Now factor and get all zeros of function.

Try: Factor completely given that is a factor.

Ex #2 Given one ______of a polynomial function, find the other zeros.

Hint: A zero is a solution of the polynomial. Use synthetic division with this number to break down polynomial until you can factor. Set each factor equal to zero to solve for zeros.

One zero of is . What is another zero?

3 1 -2 -23 60

Try: One zero of is 4. What is another zero?

Day 31: 5-6 Find rational zeros

Rational Zero Theorem: To find all possible zeros of a function:
___all factors of constant term____
all factors of leading coefficient

List possible zeros of f using rational zero theorem.

To find the zeros using the rational zero theorem:
Find all possible zeros and then test them one at a time using synthetic division. If there is no remainder then it is a solution.

Find all real zeros of

Try: Find all the real zeros of

Try: Find all the zeros of

5-7 Apply the fundamental theorem of Algebra

The Fundamental Theorem of Algebra:
If f(x) is a polynomial of degree n where n>0, then the equation f(x)=0 has at least one solution that is a complex number.
If f(x) is a polynomial of degree n where n>0 , then it has n solutions including repeats.

How many solutions does this equation have?

1. 2.

Find all the zeros of a polynomial function.

1. Find all possible zeros. Test using synthetic division.

2. When you can no longer factor use quadratic formula to get imaginary solutions.

Complex conjugates theorem: When you find an imaginary solution to the function there is also a complex conjugate solution to the function.
Irrational conjugates theorem: When you find a irrational solution to a function there is also a irrational conjugate solution to the function

If the function has a solution of then it also has a solution of______.

If a function has a solution of then it also has a solution of ______.

Use zeros to write a polynomial function

write the factored form of the solutions
Regroup when have a radical of imaginary number to get rid of radical or imaginary part
Multiply and expand the binomial
Combine like terms

Write a polynomial function of least degree that has rational coefficients, a leading coefficient of 1, and 3 and as zeros.

1. Since have a solution that is irrational must also have irrational conjugate as a solution.

2. Since 3 , , are solutions – make them into factored form:

3. Now re-group irrational factors

4. Now foil the brackets together.

Try: Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and 2 and .

Day 32: 5-8 Analyze graphs of polynomial functions

Zero: k is a solution to a function
Factor: (x – k) is a factor of the function
Solution: k is a solution of the function
x-intercept: if k is a real number, k is an intercept of the graph of the polynomial function. The graph passes through k on the x-axis.
If the function is in factored form – you can use x-intercepts to graph the function

1. Graph the function

Plot the intercepts
Plot some points between intercepts
Determine end behavior and draw ends of graph correctly

2. Try: Graph the function

Turning Points:
Local maximum: the y-coordinate of a turning point where it is higher than all nearby points
Local minimum: the y-coordinate of a turning point where it is lower than all nearby
points
The graph of every polynomial function of degree n has at most n -1 turning points.
If a polynomial function has n distinct real zeros, then its graph has exactly n – 1 turning points.

Graph the function. Identify the x-intercepts and the points where the local maximums and local minimums occur.

Use the graph to estimate the location of the local maximum(s) and the local minimum(s) if any. State the least degree of the polynomial.

Use the graphto estimate the location of the local maximum(s) and the local minimum(s) if any. State the least degree of the polynomial.

5-9 Write polynomial functions

Write a cubic function:

Use three given x-intercepts to write the function in factored form.
Find the value of a by substituting the coordinates of the fourth point

Read x-intercepts off graph:

Write function as factors:

Solve for a

Re-write function in standard form with the “a” value.

Try: Write a cubic function whose graph passes through the points: (−2, 0), (−1, 0), (0, −8), (2, 0).