C HAPTER 4 Polynomial and Rational Functions

S ECTION 4.1 Polynomial Functions Determine roots of polynomial equations Apply the Fundamental Theorem of Algebra

P OLYNOMIAL IN ONE VARIABLE A polynomial in one variable x, is an expression of the form a 0 x n + a 1 x n-1 +….+ a n-1 x + a n x. The coefficients a 0, a 1,a 2,…, a n, represent complex numbers (real or imaginary), a 0 is not zero and n represents a nonnegative integer. Example: 1000x x x 5 Degree The greatest exponent of its variable Leading Coefficient The coefficient with the greatest exponent 1000x x x 5 Degree – 18, Leading Coefficient

P OLYNOMIALS Polynomial Function If a function is defined by a polynomial in one variable with real coefficients F(x) =1000x x x 5 Zeros The values of x for a polynomial function where f(x) = 0. Also known as the x-intercepts.

P OLYNOMIALS Consider f(x) = x x x – 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) = 3x 4 – x 3 + x 2 + 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

P OLYNOMIALS Polynomial Equation A polynomial that is set equal to zero Root The solution for a polynomial equation Zero and Root are often used interchangeably but technically, you find the zero of a function and the root of an equation. Can be an imaginary number Complex Numbers Any number that can be written in the form a + b i, where a and b are real numbers and i is the imaginary unit Pure Imaginary Numbers The complex number a + b i when a = 0 and b does not equal 0 and i is the imaginary unit

P OLYNOMIALS Fundamental Theorem of Algebra Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers. Corollary to the Fundamental Theorem of Algebra States that the degree of a polynomial indicates the number of possible roots of a polynomial equation

P OLYNOMIAL G RAPHS Graphs on pg 207 Positive leading coefficients and degree greater than 0 (Top Section) Shows maximum number of times the graph of each type of polynomial may cross the x-axis General shape of a third degree function and a fourth-degree function. (Bottom Section) The graph of a polynomial function with odd degree MUST cross the x-axis AT LEAST ONCE The graph of a function with even degree MAY or MAY NOT cross the x-axis; if it does it will an even number of times

P OLYNOMIAL GRAPHS Each x-intercept represents a real root of the polynomial equation If a and b are roots of the equation, then using the corollary to the Fundamental Theorem of Algebra, we can find the polynomial equation Set equation up starting with (x-a)(x- b)=0.

S ECTION 4.2 Quadratic Equations Solve quadratic equations Use the discriminate to describe the roots of quadratic equations

Q UADRATIC E QUATIONS A polynomial equation with a degree of two. Ways to Solve Quadratic Equations: Graph Factor Completing the square Used to create a perfect square trinomial Useful when the equation can’t be factored (x + b) 2 = x 2 + 2bx + b 2 Given first and middle term, find last Square of half the coefficient of the middle term; only works with the coefficient of the first term is 1

Q UADRATIC E QUATIONS Ex. x 2 -6x -16 Graph Look at x intercepts Factor (x+2) and (x-8) Set equal to zero, x = -2, 8 Completing the square (x-3) 2 =25 x = -2, 8 Ex. 3x 2 +7x + 7 Graph Look at x intercepts  No x intercepts; roots are imaginary numbers Completing the square (x+ 7/6) 2 =-35/36 x = -7/6+/-i(35) 1/2 /6

Q UADRATIC E QUATIONS Quadratic Formula Discriminant -Tells the nature of the roots of a quadratic equation or the zeros of the related quadratic function

Q UADRATIC E QUATIONS b 2 -4ac >0 Two distinct real roots b 2 -4ac=0 Exactly one real root (actually a double root) b 2 -4ac<0 No real roots (Two distinct imaginary roots)

Q UADRATIC E QUATIONS Find the discriminant of x 2 -4x +15 and describe the nature of the roots of the equation. Then find the roots. Discriminant = -44; D<0  no real roots Roots: 2-i(11) ½ and 2+i(11) ½ Conjugates Suppose a and b are real numbers with b not equal to 0. If a + bi is a root of a polynomial equation with real coefficients, then a – bi is also a root of the equation. a + bi and a – bi are conjugate pairs What are other examples? i and –i; -1 + i and -1 – i

Q UADRATIC E QUATIONS Solve 6x 2 + x +2 by using graphing, factoring, completing the square, and the quadratic equation. Graphing The graph does not touch the x-axis  no real roots for the equation, can’t determine roots from graph Factoring No real roots, factoring can’t be solved Completing the square (x + 1/12) 2 = -47/144 Roots: -1+/-i(47) 1/2 /12 Quadratic Equation A = 6, b = 1, c = 2 X= -1+/-i(47) 1/2 /12

S ECTION 4.3 The Remainder and Factor Theorems Find the factors of polynomials using the Remainder and Factor Theorem

T HE R EMAINDER AND F ACTOR T HEOREMS Quotient DivisorDividend Remainder

T HE R EMAINDER AND F ACTOR T HEOREMS 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)

T HE R EMAINDER AND F ACTOR T HEOREMS What is 2x 2 + 3x -8 divided by x -2? Solve using long division Solve using synthetic 2x /(x-2) Divide x 3 – x 2 +2 by x +1? Solve using long division Solve using synthetic x 2 -2x + 2

T HE R EMAINDER AND F ACTOR T HEOREMS 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: 2x 3 – 3x 2 +x divided by x-1 Is the quotient a factor and/or a depressed polynomial? Yes it is both, 2x 2 -x

T HE R EMAINDER AND F ACTOR T HEOREMS Determine the binomial factors of x 3 – 7x +6 using synthetic division R R Factors are: X+3, X-1, X-2

T HE R EMAINDER AND F ACTOR T HEOREMS Determine the binomial factors of x 3 – 7x +6 using the Factor Theorem Test values F(x) = x 3 – 7x +6; Test -1 No because = 12 F(x) = x 3 – 7x +6; Test 1 Yes works because = 0, then find depressed polynomial Depressed polynomial is x 2 + x -6 Now Factor depressed polynomial to get other factors Factors to (X-1)&(X-2) All Factors are (X+3),(X-1)&(X-2)

T HE R EMAINDER AND F ACTOR T HEOREMS Determine the binomial factors of x 3 -2x 2 -13x-10 X+1, X+2, X-5 Find the value of K so that the remainder of (x 3 + 3x 2 – 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

S ECTION 4.4 T HE R ATIONAL R OOT T HEOREM Learn how to identify all possible rational roots of a polynomial equation using the rational root theorem Determine the number of positive and negative real roots each polynomial function has

T HE R ATIONAL R OOT T HEOREM Let a 0 x n + a 1 x n-1 + …+ a n-1 x + a n =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 a n and q is a factor of a 0. P is a factor of the last coefficient and Q is a factor of the first coefficient

T HE R ATIONAL R OOT T HEOREM List the possible roots of 6x 3 +11x 2 -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

T HE R ATIONAL R OOT T HEOREM Integral Root Theorem Let x n + a 1 x n-1 + …+ a n-1 x + a n =0 represent a polynomial equation that has a leading coefficient of 1, integeral coefficients, and a n can’t equal 0. Any rational roots of this equation must be integral factors of a n. Roots have to be a factor of a n, the last coefficient

T HE R ATIONAL R OOT T HEOREM Find the roots of x 3 +8x 2 +16x+5=0 How many roots are there? 3 What do they have to be factors of according to the integral root theorem? 5 Possible roots: +/-5 and +/-1 Do synthetic division with these roots to check which is a factor. (IE no remainder) Try 5 Doesn’t work, remainder of 410 Try -5 Works, no remainder Factor or use quadratic formula to find the roots of the depressed polynomial. Roots: -5, -3-(5) 1/2 /2, -3+(5) 1/2 /2

T HE R ATIONAL R OOT T HEOREM Descartes’ Rule of Signs Suppose P(x) is a polynomial whose terms are arranged in descending powers of the variable. Then the number of POSITIVE real zeros of P(x) is the same as the number of changes in sign of the coefficients of the terms or is less than this by an even number. The number of NEGATIVE real zeros of P(x) is the same as the number of changes in sign of the coefficients of the terms in P(- x) or less than this number by an even number

T HE R ATIONAL R OOT T HEOREM Find the number of possible positive real zeros and the number of possible negative real zeros for f(x) = 2x 5 +3x 4 - 6x 3 +6x 2 -8x+3 Positive Real Zeros: 4 Changes, 4, 2, or 0 possible positive real zeros Negative Real Zeros: F(-x) = -2x 5 +3x 4 +6x 3 +6x 2 +8x+3 One change, 1 possible negative real zero Find Possible zeros Possible Values of P: +/-1, +/-3 Possible Values of Q: +/-1, +/-2 Possible Values of P/Q: +/-1, +/-3, +/-1/2, +/-3/2 Test using synthetic division or graphing Rational Roots = -3, ½, 1

S ECTION 4.5 L OCATING Z EROS OF A POLYNOMIAL F UNCTION Learn to approximate the real zeros of a polynomial function

L OCATING Z EROS OF A POLYNOMIAL F UNCTION Location Principle Suppose y = f(x) represents a polynomial function with real coefficients. If a and b are two numbers with f(a) negative and f(b) positive, the functions has at least one zero between a and b. IE the answer of the equation at that root or the remainder changes signs between two roots.

L OCATING Z EROS OF A POLYNOMIAL F UNCTION Determine between which consecutive integers the real zeros of F(x) = x 3 – 4x 2 – 2x + 8 are located. Method 1: Synthetic Division Test (-3, 5) Method 2: Graphing Calculator Use Table Function There is a zero at 4, and between -2 and -1, and between 1 and 2.

L OCATING Z EROS OF A POLYNOMIAL F UNCTION Approximate the real zeros of f(x) = 12x 3 -19x 2 -x+6 to the nearest tenth. How many zeros? 3 How many positive? 2 or 0 How many negative? -12x 3 -19x 2 +x+6 1 Use graphing calculator Table to see where zeros fall Between -1 and 0, between 0 and 1, and between 1 and 2. Use graphing calculator TableSet to change delta from 1 to.1 to better see where 0’s fall Use graph to trace to see 0’s Zeros are at about -.5,.7, 1.4

L OCATING Z EROS OF A POLYNOMIAL F UNCTION Upper Bound Theorem Suppose c is a positive real number and P(x) is divided by x – c. If the resulting quotient and remainder have no change in sign, then P(x) has no real zero greater than c. Thus c is an upper bound of the zeros of P(x). Helps to determine if you have found all real zeros An integer greater than or equal to the greatest real zero Lower Bound Theorem If c is an upper bound of the zeros of P(-x), then –c is a lower bound of the zeros of P(x) An integer less than or equal to the least real zero.

L OCATING Z EROS OF A POLYNOMIAL F UNCTION Find the upper and lower bound of the zeros of f(x) = x 3 + 3x 2 -5x-10 Find real zeros: -3.6, -1.4, 2 Interval of upper and lower bound? -4<=x<=2 Find the upper and lower bound interval for f(x) = 6x 3 -7x 2 -14x <=x<=3

S ECTION 4.6 R ATIONAL E QUATIONS AND P ARTIAL F RACTIONS Learn how to solve rational equations and inequalities. Learn how to decompose a fraction into partial fractions

S ECTION 4.6 R ATIONAL E QUATIONS AND P ARTIAL F RACTIONS Rational Equation An equation with one or more rational expressions What is a rational expression? The quotient of two polynomials in the form f(x)=g(x)/h(x), where h(x) does not equal 0 How do you solve rational equations? Multiply each side by the

R ATIONAL E QUATIONS AND P ARTIAL F RACTIONS Example 1: Solve a 2 -5 = a 2 +a+2 a 2 -1 a+1 What is the LCD? a 2 -1 What do we get for a? a = 3 or -1 Can both of these be our answers? a can only be 3 because if we plug in -1 to our original equation we get a denominator of 0. Example 2: Solve X – 2 = 20. X + 4 x – 1 x2 + 3x - 4 What is the LCD? x 2 + 3x – 4 which factors to (x-1) * (x+4) What is the answer? 7

R ATIONAL E QUATIONS AND P ARTIAL F RACTIONS Example 1: Decompose 8y + 7 into partial fractions. y 2 + y - 2 First Factor Denominator (y-1) *(y+2) Then split into two fractions on other side of equals 8y + 7 = A + B y 2 + y – 2 (y-1) (y+2) Multiply each piece by LCD to get rid of fractions 8y + 7 = A(y+2) + B(y-1) Eliminate B by plugging in 1 for y Solve for A A = 3 Eliminate A by plugging in 2 for y Solve for B B = 3 Re-write fractions by plugging in values found for A and B 8y + 7 = y 2 + y – 2 (y-1) (y+2) Check to see if the sum of the two fractions equal the original.

R ATIONAL E QUATIONS AND P ARTIAL F RACTIONS Example 2: Decompose 6x - 2 into partial fractionsx 2 -3x – x+2 x-5

R ATIONAL E QUATIONS AND P ARTIAL F RACTIONS Rational Inequalities Same as equations but with inequality sign Example 1: (x-2)(x-1) < 0 (x-3)(x-4) 2 For what values is our domain undefined? 3 and 4 What values make this 0? 2 and 1 Plot these points on a number line with dashes at the above values What happens to our number line? Splits into intervals Test each interval to see if our inequality is true or false Works for intervals x <1 and 2<x<3 Show Solution on number line

R ATIONAL E QUATIONS AND P ARTIAL F RACTIONS Example 2: > 3 3a 6a 4 Solve for a first, by multiplying by LCD of 12a. A = 2 What is the zero? 2 What is the excluded value? 0 Test intervals Works for intervals 0<a<2 Show Solution on number line

S ECTION 4.7 R ADICAL E QUATIONS AND I NEQUALITIES Learn how to solve equations and inequalities with radicals involved.

R ADICAL E QUATIONS AND I NEQUALITIES Radical Equations Equations in which radical expressions include variables Extraneous Solutions Solutions that do not satisfy the original equation Check all solutions back into original equation in order to exclude those that don’t work

R ADICAL E QUATIONS AND I NEQUALITIES Example: x = √x+7) +5 Solve for X x = 9 and x = 2 Check that neither are extraneous solutions Only 9 works, Answer: x=9 Example 2: 4 = 3 √ x+2)+8 Solve for X x = -66 Check Works, Answer: x=-66 Example: √ x+1) = 1 + √ 2x-12) X = 8

R ADICAL E QUATIONS AND I NEQUALITIES Radical Inequalities Same as equations but with inequality signs Example: √ 4x+5) <10 Solve for X X<23.75 Must also find the lower bound to make √ 4x+5) a real number. Set √ 4x+5) =0 and solve X>-1.25 Solution is -1.25< X<23.75 Check by testing intervals Graph intervals on number line Example: √ 6x-5) > 4 X > 7/2