HA2 Ch. 5 ReviewHA2 Ch. 5 Review

PolynomialsPolynomials

AndAnd

Polynomial FunctionsPolynomial Functions

VocabularyVocabulary

End behavior P.282End behavior P.282 Monomial P.280Monomial P.280 Multiplicity P. 291Multiplicity P. 291 Polynomial Function P.280Polynomial Function P.280 Relative Maximum/Min. P.291Relative Maximum/Min. P.291 Standard Form of a Poly. FunctionStandard Form of a Poly. Function Synthetic Division P.306Synthetic Division P.306 Turning Point P. 282Turning Point P. 282

5-1 Polynomial Functions5-1 Polynomial Functions

Degree of a PolynomialDegree of a Polynomial – highest – highest exponentexponent

Standard FormStandard Form – in descending order – in descending order Define a polynomial by degree and Define a polynomial by degree and

by number of termsby number of terms – See green – See green table on P. 281table on P. 281

Maximum # of Turning points:Maximum # of Turning points: n-1 n-1 End behaviorEnd behavior – the far left and far – the far left and far

right of the graph right of the graph

End Behavior P. 282 End Behavior P. 282

Think in terms of a parabolaThink in terms of a parabola If even, and a +If even, and a + Then end behavior upward facingThen end behavior upward facing

End BehaviorEnd Behavior

Think in terms of a parabolaThink in terms of a parabola If even, and a negativeIf even, and a negative Then end behavior downward facingThen end behavior downward facing

End Behavior End Behavior

Think in terms of a parabolaThink in terms of a parabola If If ODDODD, and a +, and a + Then Then rightright end behavior upward end behavior upward

facing, facing, leftleft is down is down

End BehaviorEnd Behavior

Think in terms of a parabolaThink in terms of a parabola If If ODDODD, and a negative, and a negative Then Then right right end behavior downward end behavior downward

facing, facing, left left is upis up

Graphing Polynomial FunctionsGraphing Polynomial Functions

Step 1 – Find zeros and points in Step 1 – Find zeros and points in between.between.

Step 2 – “Sketch” graphStep 2 – “Sketch” graph Step 3 – Use end behavior to checkStep 3 – Use end behavior to check Try to graph: y = 3x - xTry to graph: y = 3x - x³³ Factored: 0 = x (3-xFactored: 0 = x (3-x²²))

x = 0, ±x = 0, ±√3√3

GraphGraph

AssessAssess

What is the end behavior and What is the end behavior and maximum amount of turning points maximum amount of turning points in:in:

(1.) y = -2x² - 3x + 3(1.) y = -2x² - 3x + 3

(2.) y = x³ + x + 3(2.) y = x³ + x + 3

(1.) down and down, max 1 turning (1.) down and down, max 1 turning pointpoint

(2.) down and up, max two tp(2.) down and up, max two tp

5-2 Polynomials, Linear Factors, 5-2 Polynomials, Linear Factors, and Zerosand Zeros

Factoring Polynomials:Factoring Polynomials: GCFGCF Patterns: Diff of Squares, Perf. Sq. TrinomialPatterns: Diff of Squares, Perf. Sq. Trinomial X-Method, Reverse Foil, Guess and CheckX-Method, Reverse Foil, Guess and Check Set factors equal to zero and solve.Set factors equal to zero and solve.

If those methods don’t work, then If those methods don’t work, then use Quadratic Formula to solve:use Quadratic Formula to solve:• X =X =

Multiplicity – Factor repeatsMultiplicity – Factor repeats• What are the zeros of What are the zeros of

f (x) = xf (x) = x⁴ - 2x³ - 8x² and what are their ⁴ - 2x³ - 8x² and what are their mult. ?mult. ?

• = x (x² - 2x – 8)= x (x² - 2x – 8)• = x (x + 2)(x – 4)= x (x + 2)(x – 4)• x = 0 (x2), -2, and 4x = 0 (x2), -2, and 4

Writing a FunctionWriting a Function

What is a cubic polynomial function What is a cubic polynomial function in standard form with zeros 4, -1, and in standard form with zeros 4, -1, and 2?2?

y = (x – 4)(x + 1)(x – 2)y = (x – 4)(x + 1)(x – 2) y = (x² - 3x -4)(x – 2)y = (x² - 3x -4)(x – 2) y = x³ - 5x² + 2x + 8y = x³ - 5x² + 2x + 8

5-3 Solving Polynomial 5-3 Solving Polynomial EquationsEquations

Two more patterns for factoring:Two more patterns for factoring: Sum/Diff of Cubes:Sum/Diff of Cubes:

• a³ + b³ = (a + b)(a² - ab +b)a³ + b³ = (a + b)(a² - ab +b)• a³ – b³ = (a - b)(a² + ab +b)a³ – b³ = (a - b)(a² + ab +b)

Factor completely: x³ - 27Factor completely: x³ - 27 = (x – 3)(x + 3x + 9)= (x – 3)(x + 3x + 9) What are the real/imaginary solutions?What are the real/imaginary solutions? Solve: x = 3, Solve: x = 3, ± ±

5-3 Word Problem5-3 Word Problem

The width of a box is 2 m less than The width of a box is 2 m less than the length. The height is 1 m less the length. The height is 1 m less than the length. The volume is 60 than the length. The volume is 60 m³. What is the length of the box?m³. What is the length of the box?

5-4 Dividing Polynomials5-4 Dividing Polynomials

Remember to write in standard form Remember to write in standard form and put a zero place holder.and put a zero place holder.

Use long division to determine if Use long division to determine if

(x – 2) is a factor of x - 32.(x – 2) is a factor of x - 32. Remember in synthetic division if you Remember in synthetic division if you

have a fraction to divide your answer have a fraction to divide your answer by denominator.by denominator.