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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: axn Up and Up a Positive n Even Down and Up a Positive n Odd Down and Down a Negative n Even 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) 2m2 – 3+7m 3) x2 – x4 +2x2 Determine the end behavior of the graph of each polynomial function. 4) y=-3x + 6x2–1 5) y=-3 – 6x5–9x8 6) y=-x3–x2+3 Determine the degree of the polynomial: 7) x y -2 -15 -1 -9 0 -9 1 -9 2 -3 Up and Down a Negative n Odd 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) x3+7x2+10x 2) x3-81x 3) x3+8x2+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 1 7) x=2, -1, 5, 3 5.3 Solving Polynomial Equations 6x3+15x2+9x 8x2+10x-3 4x2+12x+9 x2-36 2x3+2x2+3x+3 x3-27 Examples: Leave 2-3+ lines for each problem Find the real or imaginary solutions of each equation by factoring. 1) 2) 3) 4) 5) 6) 7) 8) 2x3 – 5x2 = 3x 3x4 + 12x2 = 6x3 (x2 – 1)(x2 + 4) = 0 x5 + 4x3 = 5x4 – 2x3 x4 = 16 x3 = 8x – 2x3 x3 = 1 x(x2 + 8) = 8(x + 1) DO THESE EXAMPLES! BEFORE CLASS! 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) (x2– 3x – 40) ÷ (x+5) 2) (x3+ 3x2 –x +2) ÷ (x-1) Divide using synthetic division: 3) (2x2+7x+11) ÷ (x+2) 4) (x3+5x2+11x + 15) ÷ (x+3) 5) (6x2+7x+2) ÷ (2x+1) Use synthetic division and the Remainder Theorem to find P(a). 6) P(x) = x3+4x2+4x; a= -2 7) P(x) = x3+7x2+4x; a= -2 5.5 Notes: Theorems About Roots of Polynomial Equations Rational Root Theorem: factor of constant term factor of leading coefficient *To find actual roots, in root for x value; if it by =0,lead thencoefficient’s it’s a root) factors) ± use Remainder Theorem (plug *(divide constant’s 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) x3 – 4x + 1 = 0 2) 2x3 – 5x + 4 =0 3) 2x3 + x2 – 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 5.6 The Fundamental Theorem of Algebra The degree of a polynomial tells you how many roots it has. (Ex: x2=2 roots, x5=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 0. 4. 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) x3+2x2-5=0 2) x5-17=0 3) x14-x7+x3-2=0 Without using a calculator, find all the roots of each equation. 4) 5) 6) 7) y=x3-4x2+9x-36 x5-x4-3x3+3x2-4x+4=0 y=x4-x3-5x2-x-6 x5-3x3-4x=0 5.7 The Binomial Theorem Expand: multiplying an expression (normally using the binomial theorem) Pascal’s Triangle: a formula for expanding a binomial: Row Power 0 (a+b)0 1 (a+b)1 2 (a+b)2 3 (a+b)3 Expanded Form 1 1a1+1b1 1a2+2a1b1+1b2 1a3+3 a2b1+3a1b2+1b3 …etc… Coefficients of expanded form match Pascal’s Triangle *exponents always add to power you are expanding to Examples: (Leave 2 lines each) 1) (x+3)4 2) (3-z)3 3) (x+y)5 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=x3. 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