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Chapter 7 POLYNOMIAL FUNCTIONS Polynomial in one variable A polynomial in one variable x, is an expression of the form a0xn + a1xn-1 +….+ an-1x + anx. The coefficients a0, a1,a2,…, an, represent complex numbers (real or imaginary), a0 is not zero and n represents a nonnegative integer. Example: 1000x18 + 500x10 + 250x5 Degree The greatest exponent of its variable Also tells the number of zeros or roots Leading Coefficient The coefficient with the greatest exponent 1000x18 + 500x10 + 250x5 Degree – 18, Leading Coefficient - 1000 Polynomials Consider f(x) = x3 + -6x2 + 10x – 8 State the degree and leading coefficient. Degree of 3 and leading coefficient of 1 Determine whether 4 is a zero of f(x). Evaluate f(4) Yes it is a zero. Example f(x) = 3x4 – x3 + x2 + x – 1 State the degree and leading coefficient Degree 4, leading coefficient of 3 Determine whether -2 is a zero of f(x) No it is not a zero of the polynomial Common polynomial functions Type Constant Linear Quadratic Cubic Quartic Quintic Degree Examples 0 9 1 x-2 2 3x2-3x+4 3 3x3-6x2-3x+4 4 x4-2x3-5x2+4x-3 5 x5+3x4-7x3-x2-x+2 Evaluating Functions F(x)=3x2-3x+1 Find values of F(0)= 3(0)2-3(0)+1=1 F(1)= 3(1)2-3(1)+1 1 F(-1)= 3(-1)2-3(-1)+1 7 Evaluating polynomials f(x) = 3x² - 3x + 1 Evaluate f(a) p(x) = x³ +4x² - 5x Evaluate p(a²) q(x) = x² + 3x =4 Evaluate q(a + 1) Graphs of Polynomials End Behavior What happens to f(x), or y, as x approaches infinity Even degree function “ends” in same direction May/may not cross x-axis If it does, it crosses it an EVEN amount of times Odd degree function “ends” in opposite direction Always crosses x-axis at least once Even Degree Odd Degree F(x) = x2 F(x) = x2 F(x) = x3 F(x) =- x3 Determine the end behavior Determine whether it is odd or even State the number of real and complex zeros 7.2 Graph using calculator Relative minimum – lowest turning point in an interval Absolute Maximum-he least value that a function assumes over its domain Relative maximum – highest turning point in an interval Absolute Maximum- the greatest value that a function assumes over its domain *Find by tracing on calculator 7.3 Solve using quadratic form A polynomial can be written in quadratic form if you square the middle variable and it equals the first variable U Substitution To write an expression in quadratic form: 1. Let u = middle variable 2. Re-write the first variable as u² 3. Factor 4. Replace original variable and solve 7.4 Remainder Theorem Remainder Theorem If a polynomial P(x) is divided by x – r, the remainder is a constant P(r), and P(x) =(x-r) * Q(x) + P(r), where Q(x) is a polynomial with degree one less than the degree of P(x) Use synthetic substitution If you divide f(x) by x – a, the remainder is f(a) The Remainder and Factor Theorems Quotient Divisor Dividend Remainder The Remainder and Factor Theorems Factor Theorem The binomial x – r is a factor of the polynomial P(x) if and only if P(r) = 0. IE. No remainder Depressed Polynomial The quotient when a polynomial is divided by one of its binomial factors x – r, Ex: 2x3 – 3x2 +x divided by x-1 Is the quotient a factor and/or a depressed polynomial? Yes it is both, 2x2 -x Factor Theorem Do synthetic substitution If f(a) has a remainder of 0, then x – a is a factor of the polynomial Factor the depressed polynomial to find the remaining factors The Remainder and Factor Theorems What is 2x2 + 3x -8 divided by x -2? Solve using long division Solve using synthetic 2x + 7 + 6/(x-2) Divide x3 – x2 +2 by x +1? Solve using long division Solve using synthetic x2 -2x + 2 Summary of Roots, Zeros, Solutions, Factors etc. P(X) Polynomial Function C is a ZERO (x-c) is a factor P(x)=0Polunomial Equation C is a Root or Solution (x-c) is a factor The Rational Root Theorem Let a0xn + a1xn-1 + …+ an-1x + an =0 represent a polynomial equation of degree n with integral coefficients. If a rational number p/q, where p and q have no common factors, is a root of the equation, then p is a factor of an and q is a factor of a0. P is a factor of the last coefficient and Q is a factor of the first coefficient P/Q are possible roots of polynomial 7.6 Rational Root Theorem To find all possible zeros: 1. Look at first (Q) and last (P) coefficient 2. List all (±) factors of last coefficient 3. List all of these again but divide each last factor (P) by every last factor (Q) 4. Ignore repeats The Rational Root Theorem List the possible roots of 6x3+11x2-3x-2=0 P must be a factor of 2 Q must be a factor of 6 Possible Values of P: +/-1, +/-2 Possible Values of Q: +/-1, +/-2, +/-3, +/-6 Possible rational roots, p/q : +/-1, +/-2, +/-1/2, +/-1/3, +/-1/6, +/-2/3 Use graphing to narrow down the possibilities Find zero at X = -2 Check using synthetic, then factor the depressed polynomial to get roots X = -2, -1/3, 1/2 The Remainder and Factor Theorems Determine the binomial factors of x3 -2x2-13x- 10 X+1, X+2, X-5 Find the value of K so that the remainder of (x3 + 3x2 – kx – 24) divided by (x + 3) is 0. Set dividend equal to 0, plug in -3 for X, and then solve for K K=8 Check using synthetic division To find all zeros… 1. Find one the hard way 2. Use synthetic substitution to find a depressed polynomial 3. Factor answer and set each equal to zero to find the other zeros Hints for finding zeros Try to trace all on calculator first Find one on calc, use this for synthetic substitution If you know that one zero is a complex #, another zero is ALWAYS its conjugate (find the other by graphing or rational zero theorem) Operations with Functions Sum: (f+g)(x)=f(x) + g(x) Difference: (f-g)(x)=f(x) - g(x) Product: (f*g)(x)=f(x) * g(x) Quotient: f =f(x) / g(x) x g Composition of Functions (f o g)(x) means f[g(x)] f[g(x)] means to substitute the function g(x) wherever you see an x in f(x) Inverse relations To find the inverse of a relation, flip-flop x and y in each ordered pair To find the inverse, f ˉ¹(x), of a function: 1. Replace f(x) with y 2. Interchange x and y 3. Solve for y 4. Replace y with f ˉ¹(x) Inverse Functions Ex. F(x) = (x + 3)2 - 5 F(x)-1 = -3 + -(x+5) 1/2 Ex. F(x) = 1/(x)3 F(x)-1 =1/(x)1/3 The graphs of inverse functions are symmetric across the line y = x To determine if two functions are inverses… Method 1 Inverses if (f o g)(x) = x and (g o f)(x) = x Method 2 Each inverse is the other function