Chapter 11 Polynomial Functions 11.1 Polynomials and Polynomial Functions.
+ Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.
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Transcript of + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.
![Page 1: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/1.jpg)
+Warm Up #1
![Page 2: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/2.jpg)
+
PolynomialsUnit 6
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+6.1 - Polynomial Functions
![Page 4: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/4.jpg)
+Objectives
By the end of today, you will be able to…
Classify polynomials
Model data using polynomial functions
![Page 5: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/5.jpg)
+Vocabulary A polynomial is a monomial or the sum of
monomials.
The highest exponent of the variable determines the degree of that polynomial.
standard form of a polynomial - Ordering the terms by degree in descending order
P(x) = 2x³ - 5x² - 2x + 5
Leading Coefficient
Cubic Term
Quadratic Term
Linear Term
Constant Term
![Page 6: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/6.jpg)
+ Standard Form of a Polynomial
For example: P(x) = 2x3 – 5x2 – 2x + 5
PolynomialStandard Form
Polynomial
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+Parts of a Polynomial
P(x) = 2x3 – 5x2 – 2x + 5
Standard Form:
Leading Coefficient:
Cubic Term:
Quadratic Term:
Linear Term:
Constant Term:
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+Parts of a Polynomial
P(x) = 4x2 + 9x3 + 5 – 3xStandard Form:
Leading Coefficient:
Cubic Term:
Quadratic Term:
Linear Term:
Constant Term:
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+ Classifying Polynomials
1) By the degree of the polynomial (or the largest degree of any term of the polynomial.
Degree Name Example
0 Constant 7
1 Linear 2x + 5
2 Quadratic 2x2
3 Cubic 2x3 – 4x2 + 5x + 4
4 Quartic x4 + 3x2
5 Quintic 3x5 – 3x + 7
![Page 10: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/10.jpg)
+Graphs are based on degrees!
Constant
Linear
Quadratic Cubic
Quartic
![Page 11: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/11.jpg)
+Classifying Polynomials
We can classify polynomials in two ways:
2) By the number of terms
# of Terms Name Example
1 Monomial 3x
2 Binomials 2x2 + 5
3 Trinomial 2x3 + 3x + 4
4 Polynomial with 4 terms
2x3 – 4x2 + 5x + 4
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+Classifying Polynomials
Write each polynomial in standard form. Then classify it by degree AND number of terms.
1. -7x2 + 8x5 2. x2 + 4x + 4x3 + 4
3. 4x + 3x + x2 + 5 4. 5 – 3x
![Page 13: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/13.jpg)
+Review – Regression Models 1) Find a linear model for the data below (STAT CALC LinReg
2) Find a quadratic model for the data
(STAT CALC QuadReg)
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+ Cubic Regression
We have already discussed regression for linear functions, and quadratic functions. We can also determine the Cubic model for a given set of points using Cubic Regression.
STAT Edit
x-values in L1, y-values in L2
STAT CALC
6:CubicReg
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+ Cubic Regression
Find the cubic model for each function:
1. (-1,3), (0,0), (1,-1), (2,0)
2. (10, 0), (11,121), (12, 288), (13,507)
![Page 16: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/16.jpg)
+Picking a Model
Given Data, we need to decide which type of model is the best fit.
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+
x y0 2.82 54 66 5.58 4
Using a graphing calculator, determine whether a linear, quadratic, or cubic model best fits the values in the table.
Enter the data. Use the LinReg, QuadReg, and CubicReg options of a graphing calculator to find the best-fitting model for each polynomial classification.
Graph each model and compare.
The quadratic model appears to best fit the given values.
Linear model Quadratic model Cubic model
Comparing Models
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+
Polynomial
Models
You have already used lines and parabolas to model data. Sometimes you can fit data more closely by using a polynomial model of degree three or greater.
Using a graphing calculator, determine whether a linear model, a quadratic model, or a cubic model best fits the values in the table.
x 0 5 10 15 20
y 10.1 2.8 8.1 16.0 17.8
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+Exit Ticket1) Determine which type of model best fits the
values in the table (Linear, Quadratic, or Cubic) and find the model
2) Write 2x(3x2 + 4x +1) in standard form. Then classify it by degree and number of terms.
1) Standard Form:
2) Degree:
3) Classify by degree:
4) Number of Terms:
5) Classify by number of terms:
x -5 -1 0 1 5
y -5 -1 0 1 5
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+Coming up…
HW tonight – Worksheet 6.1
Unit 6 TEST – Wednesday, April 16th
(possibly Thursday 4/17)
Be prepared for a quiz at any time!!
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+Warm Up # 2
![Page 22: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/22.jpg)
+HW Check – 6.1 2) y = .013x3 - .174x2 + .795x + 3.125; when x = 7, y = 4.64
3) 5x + 2 ; Linear binomial
6) 5s4 – 2s + 1 ; Quartic trinomial
9) 2x2 – 1 ; Quadratic binomial
12) 3x3
15) a5 + a4 + a3 ; Quintic trinomial
18) 9c4 ; Quartic monomial
21) s2 + 2/3 ; Quadratic binomial
24) 3x + 5
25) y = .26x2 – 3.62x + 29.3 ; average benefit if 2005 is $955.82
26) y = .13x + 2.06 ; 12 days
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+6.2 - Polynomials & Linear Factors
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+ Factored Form
The Factored form of a polynomial is a polynomial broken down into all linear factors.
We can use the distributive property to go from factor form to standard form.
![Page 25: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/25.jpg)
+ Factored to Standard
Write the following polynomial in standard form:
(x+1)(x+2)(x+3)
![Page 26: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/26.jpg)
+Factored to StandardWrite the following polynomial in standard form:
(x+1)(x+1)(x+2)
![Page 27: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/27.jpg)
+Factored to Standard
Write the following polynomial in standard form:
x(x+5)2
![Page 28: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/28.jpg)
+Standard to Factored form
To Factor:
1. Factor out the GCF of all the terms
2. Factor the Quadratic
Example: 2x3 + 10x2 + 12x
![Page 29: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/29.jpg)
+Standard to Factored formWrite the following in Factored Form
3x3 – 3x2 – 36x
![Page 30: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/30.jpg)
+Standard to Factored form
Write the following in Factored Form
x3 – 36x
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+The Graph of a Cubic
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+Vocabulary
• Relative Maximum: The greatest Y-value of the
points in a region.
Relative Minimum: The least Y-value of the points in a region.
Zeros: Place where the graph crosses x-axis
y-intercept: Place where the graph crosses y-axis
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+ Relative Max and Min
f(x) = x3 +4x2 – 5x Relative min:
Relative max:
Calculator:2nd CALC Min or Max
Use a left bound and a right bound for each min or max.
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+Finding Zeros – from a graph
Locate the x-intercepts
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+Warm Up (Do on the back of your warm up sheet)
X -4 -2 0 2 4
Y 3 1 0 1 3
Graph the points below and decide which model would be best (Linear, Quadratic or Cubic).
Hint – Look at the scatterplot!
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+QUIZ Time! 20 minutes maximum!
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+To find zeros (x-intercepts) – Set each factor = 0 and solve for x.Find the Zeros of the Polynomial Function.
1. y = (x – 2)(x + 1)(x + 3)
2. y = (x – 7)(x – 5)(x – 3)
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+Writing a Polynomial Function
Give the zeros -2, 3, and -1, write a polynomial function in factored form.
Then rewrite it in standard form to classify it by degree and number of terms.
![Page 39: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/39.jpg)
+Give the zeros 5, -1, and -2, write a polynomial function. Then classify it by degree and number of terms.
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+Repeated Zeros
A repeated zero is called a MULITIPLE ZERO.
A multiple zero has a MULTIPLICITY equal to the number of times the zero repeats.
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+Find the Multiplicity of a Zero
Find any multiple zeros and their multiplicity
y = x4 + 6x3 + 8x2
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+Find the Multiplicity of a Zero
Find any multiple zeros and their multiplicity
1. y = (x – 2)(x + 1)(x + 1)2
2. y = x3 – 4x2 + 4x
![Page 43: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/43.jpg)
+Warm Up #3
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+Homework Check – 6.2
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+6.3 Dividing Polynomials
![Page 46: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/46.jpg)
+Vocabulary
Dividend: number being divided
Divisor: number you are dividing by
Quotient: number you get when you divide
Remainder: the number left over if it does not divide evenly
Factors: the DIVISOR and QUOTIENT are FACTORS if there is no remainder
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+Long Division
Divide WITHOUT a calculator!!
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+Steps for Dividing
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+Using Long Division on Polynomials
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+Divide
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+Using Long Division on Polynomials
![Page 52: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/52.jpg)
+Synthetic Division
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+Synthetic Definition
To divide by a linear factor, you can use a simplified process that is known as synthetic division. In synthetic division, you omit all variables and exponents.
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+Synthetic Division Steps:
1. Switch the sign of the constant term in the divisor. Write the coefficients of the polynomial in standard form.
2. Bring down the first coefficient.
3. Multiply the first coefficient by the new divisor.
4. Repeat step 3 until remainder is found.
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+Example
Use Synthetic division to divide
3x3 – 4x2 + 2x – 1 by x + 1
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+Example
Use Synthetic division to divide
X3 + 4x2 + x – 6 by x + 1
![Page 57: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/57.jpg)
+Check your work!
Dividend = Divisor x Quotient + Remainder
![Page 58: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/58.jpg)
+Example
Use Synthetic division to divide
X4 + 4x2 + x – 6 by x + 1
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+Example
Use Synthetic division to divide
X3 + 3x2 – x – 3 by x – 1
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+Remainder Theorem
If a polynomial P(x) is divided by (x – a), where a is a constant, then the remainder is P(a).
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+Find the remainder for
P(x) = x4 – 5x2 + 4x + 12 divided by (x + 4) using the Remainder Theorem
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+6.4 Solving Polynomials by Graphing
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+Solving by Graphing: 1st Way
Solutions are zeros on a graph
Step 1: Solve for zero on one side of the equation.
Step 2: Graph the equation
Step 3: Find the Zeros using 2nd CALC
(Find each zero individually)
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+
Step 1: Graph both sides of the equal sign as two separate equations in y1 and y2.
Use 2nd CALC Intersect to find the x values at the points of intersection
Solving by Graphing: 2nd Way
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+Solve by Graphing
x3 + 3x2 = x + 3
x3 – 4x2 – 7x = -10
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+Solve by Graphing
x3 + 6x2 + 11x + 6 = 0
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+Solving by Factoring
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+Factoring Sum and Difference
Factoring cubic equations:
Note: The second factor is prime (cannot be factored anymore)
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+ Factor:
1) x3 - 8
2) 27x3 + 1
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+You Try! Factor:
1) x3 + 64
2) 8x3 - 1
3) 8x3 - 27
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+
Solving a Polynomial Equation
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+Solving By Factoring
Remember: Once a polynomial is in factored form, we can set each factor equal to zero and solve.
4x3 – 8x2 + 4x = 0
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+Solve by factoring:
1. 2x3 + 5x2 = 7x
2. x2 – 8x + 7 = 0
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+Using the patterns to Solve
So solve cubic sum and differences use our pattern to factor then solve.
X3 – 8 = 0
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+Using the patterns to Solve
x3 – 64 = 0
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+Using the patterns to Solve
x3 + 27 = 0
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+
Factoring by Using Quadratic Form
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+Factoring by using Quadratic Formx4 – 2x2 – 8
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+Factoring by using Quadratic Formx4 + 7x2 + 6
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+Factoring by using Quadratic Formx4 – 3x2 – 10
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+Solving Using Quadratic Form
x4 – x2 = 12
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+
6.5 Theorems About Roots
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+The Degree
Remember: the degree of a polynomial is the highest exponent.
The Degree also tells us the number of Solutions (Including Real AND Imaginary)
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+Solutions/Roots
How many solutions will each equation have? What are they?
1. x3 – 6x2 – 16x = 0
2. x3 + 343 = 0
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+Solving by Graphing
Solving by Graphing ONLY works for REAL SOLUTIONS. You cannot find Imaginary solutions from a Graph.
Roots: This is another word for zeros or solutions.
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+Rational Root Theorem
If p/q is a rational root (solution) then:
p must be a factor of the constant
and
q must be a factor of the leading coefficient
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+Example
x3 – 5x2 - 2x + 24 = 0
Lets look at the graph to find the solutions
Factored (x + 2)(x – 3)(x – 4) = 0
Note: Roots are all factors of 24 (the constant term) since a = 1.
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+Example
24x3 – 22x2 - 5x + 6 = 0
Lets look at the graph to find the solutions:
Factored (x + ½ )(x – ⅔)(x – ¾ ) = 0 1,2, and 3 (the numerators) are all factors of 6 (the
constant).
2, 3, and 4 (the denominators) are all factors of 24 (the leading coefficient).
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+ 8) x3 – 5x2 + 7x – 35 = 0
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+ 10) 4x3 + 16x2 -22x -10 = 0
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+Irrational Root Theorem
Square Root Solutions come in PAIRS:
If x2 = c then x = ± √c
If √ is a solution so is -√
Imaginary Root Theorem
If a + bi is a solution, so is a – bi
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+Recall
Solve the following by taking the square root:
X2 – 49 = 0
X2 + 36 = 0
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+Using the Theorems
Given one Root, find the other root!
1. √5 2. -√6
3. 2 – i 4. 2 - √3
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+Zeros to Factors
If a is a zero, then (x – a) is a factor!!
When you have factors
(x – a)(x – b) = x2 + (a+b)x + (ab)
SUM PRODUCT
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+Examples
1. Find a 2nd degree equation with roots 2 and 3
(x - _______)(x - ______)
2. Find a 2nd degree equation with roots -1 and 6
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+Example
1. Find a 2nd degree equation with roots ±√7
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+Examples
1. Find a 2nd degree equation with roots ±2√5
2. Find a 2nd degree equation with roots ±6i
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+Examples
Find a 2nd degree equation with a root of 7 + i
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+Example
Find a 3rd degree equation with roots 4 and 3i
(x - _______)(x - ______)(x - ______)
![Page 100: + Warm Up #1. + Polynomials Unit 6 + 6.1 - Polynomial Functions.](https://reader035.fdocuments.us/reader035/viewer/2022081503/56649ef15503460f94c02693/html5/thumbnails/100.jpg)
+Example
Find a third degree polynomial equation with roots 3 and 1 + i.