Leading Term: ax n. Up and Up Down and Up Down and Down Up and Down a Positive a Positive a Negative a Negative n Even n Odd n Even n Odd

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1 5.1 Notes: Polynomial Functions monomial: a real number, variable, or product of one real number and one or more variables degree of monomial: the exponent of the variable(s) polynomial: a monomial or sum of monomials degree of polynomial: the greatest degree among its monomial terms *Constant Difference: From the y-values on a table, the differences between each term is constant (this tells you the degree) polynomial function: the degree of the polynomial function standard form: arranges the terms by degree in descending numerical order Leading Term: ax n Up and Up Down and Up Down and Down Up and Down a Positive a Positive a Negative a Negative n Even n Odd n Even n Odd Examples: (leave 2 lines) can fit 2 or 3 problems on each line Write in standard form. Classify by degree and number of terms. 1) 7x + 3x + 5 2) 2m 2 3+7m 3) x 2 x 4 +2x 2 Determine the end behavior of the graph of each polynomial function. 4) y=-3x + 6x 2 1 5) y=-3 6x 5 9x 8 6) y=-x 3 x 2 +3 Determine the degree of the polynomial: 7) x y

2 5.2 Polynomials, Linear, Factors, and Zeros Factoring: -Find GCF -Use Berry Method/unFOIL Zero Product Property: -Set all equal to zero -Solve for roots -Graph roots -Use end behavior rules to sketch function Standard Form Given Zeros: -Change signs (Example: -2 (x+2)) -Multiply using FOIL and exponent rules -Write in standard form Examples: (Leave 1 line for 1-3, 4-5 lines for 4 & 5, and about 3 lines for 6 & 7) Write in factored form. Check by multiplication. 1) x 3 +7x 2 +10x 2) x 3-81x 3) x 3 +8x 2 +16x Find the zeros of each function. Then graph the function. 4) y=(x-1)(x+2) 5) x(x+5)(x-8) Write a polynomial in standard form with the given zeros. 6) x=0, 2, 4 7) x= 1, -1, 5, 3 2

3 5.3 Solving Polynomial Equations 6x 3 +15x 2 +9x 8x 2 +10x-3 4x 2 +12x+9 x 2-36 DO THESE EXAMPLES! BEFORE CLASS! 2x 3 +2x 2 +3x+3 x 3-27 Examples: Leave 2-3+ lines for each problem Find the real or imaginary solutions of each equation by factoring. 1) 2x 3 5x 2 = 3x 2) 3x x 2 = 6x 3 3) (x 2 1)(x 2 + 4) = 0 4) x 5 + 4x 3 = 5x 4 2x 3 5) x 4 = 16 6) x 3 = 8x 2x 3 7) x 3 = 1 8) x(x 2 + 8) = 8(x + 1)

4 5.4 Notes: Dividing Polynomials Long Division: -Set up Problem with binomial on outside & polynomial on inside -Divide, multiply, subtract like usual *May get a remainder *If no remainder, that binomial is a factor of the polynomial Synthetic Division: -Solve for x of binomial (normally just change sign) -Write coefficients of polynomial in descending order (0s for placeholders) -Bring down first term -Multiply answer by divisor, then rewrite & add to next coefficient -Repeat; last number is remainder (if 0, then binomial is a factor) Remainder Theorem: P(x) is a polynomial, then P(a) is the remainder. *Plug in and evaluate function to check remainder* Examples: (Leave 5-6 lines for long division, 3 lines for synthetic) Divide using long division: 1) (x 2 3x 40) (x+5) 2) (x 3 + 3x 2 x +2) (x-1) Divide using synthetic division: 3) (2x 2 +7x+11) (x+2) 4) (x 3 +5x 2 +11x + 15) (x+3) 5) (6x 2 +7x+2) (2x+1) Use synthetic division and the Remainder Theorem to find P(a). 6) P(x) = x 3 +4x 2 +4x; a= -2 7) P(x) = x 3 +7x 2 +4x; a= -2

5 5.5 Notes: Theorems About Roots of Polynomial Equations Rational Root Theorem: factor of constant term factor of leading coefficient *To find actual roots, ± use Remainder Theorem (plug *(divide in root constant s for x value; factors if it by =0, lead then coefficient s it s a root) factors) 1) Use Table button on calculator 2) Type Equation & press enter 3) Set Start=0, Step=1, and Ask-x; Press OK 4) Type in all x values (it will do the work for you) *MUST PUT TABLE IN WORK* 5) All y values that =0 give you your x answers Conjugate Root Theorem: Rational Coefficients: If a + x is a root, then a x is a root also Real Coefficients: If a + bi is a root, then a bi is a root also *If given roots: put into factored form and multiply to get function. **remember to use both i AND i if only one is given** Examples: (Leave 3+ lines for each) Use the Rational Root Theorem to list all possible roots for each equation. Then find any actual roots. 1) x 3 4x + 1 = 0 2) 2x 3 5x + 4 =0 3) 2x 3 + x 2 7x 6=0 Write a polynomial function with rational coefficients so that P(x)=0 has the given roots. 4) 5 and 9 5) -4 and 2i

6 5.6 The Fundamental Theorem of Algebra The degree of a polynomial tells you how many roots it has. (Ex: x 2 =2 roots, x 5 =5 roots, etc.) **These roots could be real or imaginary and/or have a multiplicity greater than 1** How to find all of the roots without a calculator: 1. Try to factor (Grouping, GCF, etc.) 2a. If you can factor, factor all the way and set each factor =0 2b. If you can NOT factor, write all possible roots (± constant factors lead coefficient factors) 3. Try each root by using the remainder theorem (plug in for x) until you find a remainder of Use this factor in synthetic division (*do not change sign again; you already did) 5. Record root you used as x= 6. Factor your new polynomial after dividing 7. Use step 2a to finish problem and add to answer from step 5 Examples: Leave 1 line for problems 1-3 total & Leave 5+ entire lines for each problem 4-7 (do not break into columns) Find the number of roots: 1) x 3 +2x 2-5=0 2) x 5-17=0 3) x 14 -x 7 +x 3-2=0 Without using a calculator, find all the roots of each equation. 4) y=x 3-4x 2 +9x-36 5) x 5 -x 4-3x 3 +3x 2-4x+4=0 6) y=x 4 -x 3-5x 2 -x-6 7) x 5-3x 3-4x=0

7 5.7 The Binomial Theorem Expand: multiplying an expression (normally using the binomial theorem) Pascal s Triangle: a formula for expanding a binomial: Examples: (Leave 2 lines each) 1) (x+3) 4 2) (3-z) 3 3) (x+y) 5 Row Power Expanded Form 0 (a+b) (a+b) 1 1a 1 +1b 1 2 (a+b) 2 1a 2 +2a 1 b 1 +1b 2 3 (a+b) 3 1a 3 +3 a 2 b 1 +3a 1 b 2 +1b 3 etc Coefficients of expanded form match Pascal s Triangle *exponents always add to power you are expanding to

8 5.9 Transforming Polynomial Functions y=a(x-h) + k a=stretch (a>1) or compression (a<1) h=horizontal translation *change sign k=vertical translation *put sign in front of () if reflected Examples: (Leave 1-2 lines each) Determine the cubic function that is formed from the parent function y=x 3. 1) Vertical stretch by the factor 3 Reflection in the x-axis Vertical translation 4 units up 2) Vertical compression by factor 1/2 Reflection in the x-axis Horizontal translation 3 units to the right Vertical translation 2 units up 3) Vertical stretch by the factor 2 Horizontal translation 3 units left Vertical translation 4 units down