SPONGE (3x + 2)4 (2x – 5y)3

Dividing Polynomials

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Let’s try!  

Simple Division - dividing a polynomial by a monomial

Simplify

Simplify

COPY! How do you divide Polynomials using Long Division? Divide the leading coefficients. Multiply by the distributive property & align your terms. Subtract the polynomials. Bring down the next term. Repeat the process for all terms.

Dividend The answer = “Quotient “ Divisor

x - 8 x2/x = x -8x/x = -8 -( ) + 3x x2 - 8x - 24 -( ) - 8x - 24

 

Your turn.  

Your turn again.  

Your turn again!  

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Think Pair and Share… Your turn!  

2. Prepare for your quiz (20min only) Sponge 1. 2. Prepare for your quiz (20min only)

When you are done with your quiz….. Turn them in at my desk at get a laptop to begin on your Khan Academy.. Assignments : 1.Polynomials Intro 2. Add polynomials Intro 3. Expand Binomials

2-11-19 Synthetic Division

Sponge Find the volume.

Check !

What is Synthetic Division? http://cms.gavirtualschool.org/Shared/Math/GSEAdvan cedAlgebra/OperationsWithPolynomials/GSEAdvancedAl gebra_OperationswithPolynomials_Shared2.html#

Copy: In order to use Synthetic Division…. There must be a coefficient for every possible power of the variable The divisor must have a leading coefficient of 1.

Let’s look at how to do this using the example: YOU TRY! Recall: In order to use synthetic division these two things must happen: #1 There must be a coefficient for every possible power of the variable. #2 The divisor must have a leading coefficient of 1.

Step #1: Write the terms of the polynomial so Step #1: Write the terms of the polynomial so the degrees are in descending order.

Step #2: Write the constant a of the divisor Step #2: Write the constant a of the divisor x- a to the left and write down the coefficients. a= what goes here!! From the divisor (opposite) Coefficients

Step #3: Bring down the first coefficient, 5. Step #4: Multiply the first coefficient by r (3*5).

Step #5: After multiplying in the diagonals, add the column.

Add Multiply the diagonals, add the columns.

Add Columns Add Columns Add Columns Add Columns Step #7: Repeat the same procedure as step #6. Add Columns Add Columns Add Columns Add Columns

Step #8: Write the quotient (answer). The numbers along the bottom are coefficients of the power of x in descending order, starting with the power that is one less than that of the dividend.

The quotient is: Remember to place the remainder over the divisor.

Example 2

Check

Example 3.

Ex 3: Step#1: Powers are all accounted for and in descending order. Step#2: Identify r in the divisor. Since the divisor is x+4, r=-4 .

Test review

4 -4 20 8 -5 -1 1 -2 10 Step#3: Bring down the 1st coefficient. Step#4: Multiply and add. Step#5: Repeat. 4 -4 20 8 -5 -1 1 -2 10

Synthetic Division Can be used when dividing by x – r term, where r is a number. (4x3 + 5x2 + 8)÷(x – 2) What is x; x – 2 = 0 2 | 4 5 0 8 8 26 52 4 13 26 60 60 is the reminder; 26 is the constant, 13 the 1st degree term, 4 the 2nd degree term

Synthetic Division Can be used when dividing by x – r term, where r is a number. (4x3 + 5x2 + 8)÷(x – 2) What is x; x – 2 = 0 2 | 4 5 0 8 8 26 52 4 13 26 60 4x2 + 13x + 26 +

The Factor Theorem We use the factor theorem to help us find all “factors” of a polynomial. If the remainder equals zero, the divisor is one of the factors. To find the others you

2 7 -4 -27 -18 +2 4 22 18 36 9 2 11 18 The Factor Theorem A polynomial f(x) has a factor (x – k) if and only if f(k) = 0. Show that (x – 2) and (x + 3) are factors of f(x) = 2x4 + 7x3 – 4x2 – 27x – 18 2 7 -4 -27 -18 +2 4 22 18 36 9 2 11 18

Synthetic Division - To use synthetic division: divide a polynomial by a polynomial To use synthetic division: There must be a coefficient for every possible power of the variable. The divisor must have a leading coefficient of 1.

Step #1: Write the terms of the. polynomial so the degrees are in Step #1: Write the terms of the polynomial so the degrees are in descending order. Since the numerator does not contain all the powers of x, you must include a 0 for the

5 -4 1 6 Since the divisor is x-3, r=3 Step #2: Write the constant r of the divisor x-r to the left and write down the coefficients. 5 -4 1 6 Since the divisor is x-3, r=3

Step #3: Bring down the first coefficient, 5.

15 15 5 Step #4: Multiply the first coefficient by r, so and place under the second coefficient then add. 5 15 15

Step #5: Repeat process multiplying. the sum, 15, by r; Step #5: Repeat process multiplying the sum, 15, by r; and place this number under the next coefficient, then add. 5 15 45 41

Step #5 cont.: Repeat the same procedure. Where did 123 and 372 come from? 5 15 45 41 123 372 124 378

Step #6: Write the quotient. The numbers along the bottom are coefficients of the power of x in descending order, starting with the power that is one less than that of the dividend. 5 15 45 41 123 124 372 378

The quotient is: Remember to place the remainder over the divisor.

Ex 8: Notice the leading coefficient of the divisor is 2 not 1. We must divide everything by 2 to change the coefficient to a 1.

3

*Remember we cannot have complex fractions - we must simplify.

Ex 9: 1 Coefficients

Practice

SPONGE

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Class work practice! Handout 1

SPONGE- simplify

Function Composition UNIT 2

Relation: Any set of ___________ that has an _____________. Notes Relation: Any set of ___________ that has an _____________.   Function: A _________________ such that every single ___________ has exactly ___________ output.

Function Notation: Function notation is a way to ________________. It is pronounced ________________________. f(x) is a fancy way of writing _____ in an _______________.   Example: y = 2x + 4 is the same as ___________________

Examples 1-3  

Ex 2.  

   

Given  

Function Composition Given g(x) = 3x - 4x2 + 2, find g(5) 15 - 4(25) + 2 = 15 - 100 + 2 = -83 g(5) = -83

SPONGE 1. Given g(x) = 2x2 - 3x+5, find g(-3)

Function Composition Function Composition is just fancy substitution, very similar to what we have been doing with finding the value of a function. Key point! We will be plugging one function into the other function

The notation looks like g(f(x)) or f(g(x)). Notation Used…. The notation looks like g(f(x)) or f(g(x)). We read it ‘g of f of x’ or ‘f of g of x’

EX1. The COMPOSITION of the function f with g is Plug the second function into the first

EXAMPLE 2 Given f(x) = 2x + 2 and g(x) = 2, find f(g(x)).

Example 3 Given g(x) = x - 5 and f(x) = x + 1, find f(g(x)).

Instructional Video - RECAP https://www.youtube.com/watch?v=1x7gv3N6FD8

Example 4 YOUR TURN Find g(f(x)) Given g(x) = x - 5 and f(x) = x + 1

Example 4b YOUR TURN now find… Example 4b YOUR TURN now find…. Find f(g(x)) Given g(x) = x - 5 and f(x) = x + 1

3RD BLOCK CLASSWORK 1, 3, 5, AND 7 “COMPOSITION” You may collaborate with your peers! Must show work on a separate sheet of paper. Copy problem and show work for all 4.

Function Inverses- AFTER LUNCH

COPY: Function Inverses If given a relation set! Simply switch the x and y values! That’s the inverse when given a relation! If given an equation or function with f(x) Simply switch the x and y in the equation and then solve for the new “Y”

Function Inverse {(2, 3), (5, 0), (-2, 4), (3, 3)} Inverse Inverse = switch the x and y, (domain and range) Inverse = {(3, 2), (0, 5), (4, -2), (3, 3)}

{(4, 7), (1, 4), (9, 11), (-2, -1)} Inverse = ? Ex2. Function Inverse {(4, 7), (1, 4), (9, 11), (-2, -1)} Inverse = ? I = {(7, 4), (4, 1), (11, 9), (- 1, -2)}

Function Inverse for equations! Switch the x and y in the equation and then solve for the new “Y”

Function Inverse Ex1 given f(x) = x + 5. 1st Substitute y for f(x). y = x + 5. Then switch x and y. x = y + 5 Now solve for y. x - 5 = y (this is your inverse equation)

Final answer! So the inverse of f(x) = x + 5 is f-1(x) = x - 5 with correct notation!

Ex2. Given f(x) = 3x - 4, find its inverse (f-1(x)). y = 3x – 4 1st switch x and y. x = 3y - 4 2nd solve for y.

Your turn! Given h(x) = -3x + 9, find it’s inverse. y = -3x + 9 x = -3y + 9 x - 9 = -3y (x - 9) / -3 = f-1(x)

Classwork “Composition” Handout #2, 3 ,4 and 6 You may collaborate with your peers! Must show work on a separate sheet of paper. Copy problem and show work for all 4.

Function Inverse Given Find the inverse.

Function Inverse

Function Inverse 3x = 2y + 5 3x - 5 = 2y

Function Inverse Given f(x) = x2 - 4 y = x2 - 4 x = y2 - 4 x + 4 = y2

Function Inverse x + 4 = y2