Polynomials Chapter 6

6.1 - Polynomial Functions

Objectives By the end of today, you will be able to… Classify polynomials Model data using polynomial functions

http://www.youtube.com/watch?v=udS-OcNtSWo

Vocabulary A polynomial is a monomial or the sum of monomials. The exponent of the variable in a term determines the degree of that polynomial. Ordering the terms by descending order by degree. This order demonstrates the standard form of a polynomial. P(x) = 2x³ - 5x² - 2x + 5 Leading Coefficient Constant Term Cubic Term Quadratic Term Linear Term

Standard Form of a Polynomial For example: P(x) = 2x3 – 5x2 – 2x + 5 Polynomial Standard Form Polynomial

Parts of a Polynomial P(x) = 2x3 – 5x2 – 2x + 5 Leading Coefficient: Cubic Term: Quadratic Term: Linear Term: Constant Term:

Parts of a Polynomial P(x) = 4x2 + 9x3 + 5 – 3x Leading Coefficient: Cubic Term: Quadratic Term: Linear Term: Constant Tem:

Classifying Polynomials We can classify polynomials in two ways: 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

Classifying Polynomials 2) By the degree of the polynomial (or the largest degree of any term of the polynomial. Degree Name Example Constant 7 1 Linear 2x + 5 2 Quadratic 2x2 3 Cubic 2x3 – 4x2 + 5x + 4 4 Quartic x4 + 3x2 5 Quintic 3x5 – 3x + 7

Classifying Polynomials Write each polynomial in standard form. Then classify it by degree AND number of terms. -7x2 + 8x5 2. x2 + 4x + 4x3 + 4 3. 4x + 3x + x2 + 5 4. 5 – 3x

Cubic Regression STAT  Edit x-values in L1, y-values in L2 STAT CALC 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

Cubic Regression Find the cubic model for each function: (-1,3), (0,0), (1,-1), (2,0) (10, 0), (11,121), (12, 288), (13,507)

Picking a Model Given Data, we need to decide which type of model is the best fit.

Comparing Models Using a graphing calculator, determine whether a linear, quadratic, or cubic model best fits the values in the table. x y 0 2.8 2 5 4 6 6 5.5 8 4 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. Linear model Quadratic model Cubic model The quadratic model appears to best fit the given values.

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 5 10 15 20 y 10.1 2.8 8.1 16.0 17.8

6.2 - Polynomials & Linear Factors

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.

Factored to Standard (x+1)(x+2)(x+3) Write the following polynomial in standard form: (x+1)(x+2)(x+3)

Factored to Standard (x+1)(x+1)(x+2) Write the following polynomial in standard form: (x+1)(x+1)(x+2)

Factored to Standard x(x+5)2 Write the following polynomial in standard form: x(x+5)2

Standard to Factored form Factor out the GCF of all the terms Factor the Quadratic Example: 2x3 + 10x2 + 12x

Standard to Factored form Write the following in Factored Form 3x3 – 3x2 – 36x

Standard to Factored form Write the following in Factored Form x3 – 36x

The Graph of a Cubic

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 Vocabulary

Relative Max and Min f(x) = x3 +4x2 – 5x f(x) = -x3 – 7x2 – 18x Find the relative max and min of the following polynomials: f(x) = x3 +4x2 – 5x Relative min: Relative max: f(x) = -x3 – 7x2 – 18x (-3.2 , 24.2) (.5, -1.4) (7.1, 5) ( 3.6, -16.9) Calculator: 2nd  CALC  Min or Max Use a left bound and a right bound for each min or max.

Finding Zeros When a Polynomial is in factored form, it is easy to find the zeros, or where the graph crosses the x-axis. EX: Find the Zeros of y = (x+4)(x – 3)

Factor Theorem The Expression x – a is a linear factor of a polynomial if and only if the value a is a zero of the related polynomial function.

Find the Zeros Find the Zeros of the Polynomial Function. y = (x – 2)(x + 1)(x + 3) y = (x – 7)(x – 5)(x – 3)

Writing a Polynomial Function Give the zeros -2, 3, and -1, write a polynomial function. Then classify it by degree and number of terms. Give the zeros 5, -1, and -2, write a polynomial function. Then classify it by degree and number of terms.

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.

Find the Multiplicity of a Zero Find any multiple zeros and their multiplicity y = x4 + 6x3 + 8x2

Find the Multiplicity of a Zero Find any multiple zeros and their multiplicity y = (x – 2)(x + 1)(x + 1)2 y = x3 – 4x2 + 4x

6.3 Dividing Polynomials

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

Long Division Divide WITHOUT a calculator!! 1. 2.

Steps for Dividing

Using Long Division on Polynomials Using the same steps, divide.

Using Long Division on Polynomials Using the same steps, divide.

Using Long Division on Polynomials Using the same steps, divide.

Synthetic Division

Synthetic Division Step 1: Switch the sign of the constant term in the divisor. Write the coefficients of the polynomial in standard form. Step 2: Bring down the first coefficient. Step 3: Multiply the first coefficient by the new divisor. Step 4: Repeat step 3 until remainder is found.

Example Use Synthetic division to divide 3x3 – 4x2 + 2x – 1 by x + 1

Example Use Synthetic division to divide X3 + 4x2 + x – 6 by x + 1

Example Use Synthetic division to divide X3 + 3x2 – x – 3 by x – 1

Remainder Theorem Remainder Theorem: If a polynomial P(x) is divided by (x – a), where a is a constant, then the remainder is P(a).

Using the Remainder Theorem Find P(-4) for P(x) = x4 – 5x2 + 4x + 12.

6.4 Solving Polynomials by Graphing

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)

Solving by Graphing: 2nd Way 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

Solve by Graphing x3 + 3x2 = x + 3 x3 – 4x2 – 7x = -10

Solve by Graphing x3 + 6x2 + 11x + 6 = 0

Solving by Factoring

Factoring Sum and Difference Factoring cubic equations: Note: The second factor is prime (cannot be factored anymore)

Factor: x3 - 8 27x3 + 1

You Try! Factor: x3 + 64 8x3 - 1 8x3 - 27

Solving a Polynomial Equation

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

Solve by factoring: 1. 2x3 + 5x2 = 7x 2. x2 – 8x + 7 = 0

Using the patterns to Solve So solve cubic sum and differences use our pattern to factor then solve. X3 – 8 = 0

Using the patterns to Solve x3 – 64 = 0

Using the patterns to Solve x3 + 27 = 0

Factoring by Using Quadratic Form

Factoring by using Quadratic Form x4 – 2x2 – 8

Factoring by using Quadratic Form x4 + 7x2 + 6

Factoring by using Quadratic Form x4 – 3x2 – 10

Solving Using Quadratic Form x4 – x2 = 12

6.5 Theorems About Roots

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)

Solutions/Roots How many solutions will each equation have? What are they? x3 – 6x2 – 16x = 0 x3 + 343 = 0

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.

If p/q is a rational root (solution) then: 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

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.

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).

8) x3 – 5x2 + 7x – 35 = 0

10) 4x3 + 16x2 -22x -10 = 0

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

Recall Solve the following by taking the square root: X2 – 49 = 0 X2 + 36 = 0

Using the Theorems √5 2. -√6 3. 2 – i 4. 2 - √3 Given one Root, find the other root! √5 2. -√6 3. 2 – i 4. 2 - √3

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

Examples Find a 2nd degree equation with roots 2 and 3 (x - _______)(x - ______) 2. Find a 2nd degree equation with roots -1 and 6

Example Find a 2nd degree equation with roots ±√7

Examples Find a 2nd degree equation with roots ±2√5 Find a 2nd degree equation with roots ±6i

Examples Find a 2nd degree equation with a root of 7 + i

Example Find a 3rd degree equation with roots 4 and 3i (x - _______)(x - ______)(x - ______)

Example Find a third degree polynomial equation with roots 3 and 1 + i.