HA2 Ch. 5 Review PolynomialsAnd Polynomial Functions

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

5-1 Polynomial Functions Degree of a Polynomial – highest exponent Degree of a Polynomial – highest exponent Standard Form – in descending order Standard Form – in descending order Define a polynomial by degree and by number of terms – See green table on P. 281 Define a polynomial by degree and by number of terms – See green table on P. 281 Maximum # of Turning points: n-1 Maximum # of Turning points: n-1 End behavior – the far left and far right of the graph End behavior – the far left and far right of the graph

End Behavior P. 282 Think in terms of a parabola Think in terms of a parabola If even, and a + If even, and a + Then end behavior upward facing Then end behavior upward facing

End Behavior Think in terms of a parabola Think in terms of a parabola If even, and a negative If even, and a negative Then end behavior downward facing Then end behavior downward facing

End Behavior Think in terms of a parabola Think in terms of a parabola If ODD, and a + If ODD, and a + Then right end behavior upward facing, left is down Then right end behavior upward facing, left is down

End Behavior Think in terms of a parabola Think in terms of a parabola If ODD, and a negative If ODD, and a negative Then right end behavior downward facing, left is up Then right end behavior downward facing, left is up

Graphing Polynomial Functions Step 1 – Find zeros and points in between. Step 1 – Find zeros and points in between. Step 2 – “Sketch” graph Step 2 – “Sketch” graph Step 3 – Use end behavior to check Step 3 – Use end behavior to check Try to graph: y = 3x - x ³ Try to graph: y = 3x - x ³ Factored: 0 = x (3-x ² ) Factored: 0 = x (3-x ² ) x = 0, ± √3 x = 0, ± √3

Graph

Assess What is the end behavior and maximum amount of turning points in: What is the end behavior and maximum amount of turning points in: (1.) y = -2x² - 3x + 3 (2.) y = x³ + x + 3

(1.) down and down, max 1 turning point (1.) down and down, max 1 turning point (2.) down and up, max two tp (2.) down and up, max two tp

5-2 Polynomials, Linear Factors, and Zeros Factoring Polynomials: Factoring Polynomials: GCF GCF Patterns: Diff of Squares, Perf. Sq. Trinomial Patterns: Diff of Squares, Perf. Sq. Trinomial X-Method, Reverse Foil, Guess and Check X-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 use Quadratic Formula to solve: If those methods don’t work, then use Quadratic Formula to solve: X =X =

Multiplicity – Factor repeats Multiplicity – Factor repeats What are the zeros ofWhat are the zeros of f (x) = x ⁴ - 2x³ - 8x² and what are their mult. ? = x (x² - 2x – 8) = x (x² - 2x – 8) = x (x + 2)(x – 4) = x (x + 2)(x – 4) x = 0 (x2), -2, and 4 x = 0 (x2), -2, and 4

Writing a Function What is a cubic polynomial function in standard form with zeros 4, -1, and 2? What is a cubic polynomial function in standard form with zeros 4, -1, and 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 + 8 y = x³ - 5x² + 2x + 8

5-3 Solving Polynomial Equations 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 - 27 Factor 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 Problem The width of a box is 2 m less than the length. The height is 1 m less than the length. The volume is 60 m³. What is the length of the box? The width of a box is 2 m less than the length. The height is 1 m less than the length. The volume is 60 m³. What is the length of the box?

5-4 Dividing Polynomials Remember to write in standard form and put a zero place holder. Remember to write in standard form 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 (x – 2) is a factor of x Remember in synthetic division if you have a fraction to divide your answer by denominator. Remember in synthetic division if you have a fraction to divide your answer by denominator.