MATH 50 CHAPTER Polynomials ------- Section. 4 Degree of a Polynomial 4 Identify the following polynomials: 4 4 Descending order 7 5 4 Leading Term Leading Coefficient The constant (no variable)
LEFT AND RIGHT END BEHAVIOR Graphs of polynomials are smooth and continuous. ( no breaks, no sharp corners) Graph and see if you can see a pattern in the end behavior. 4. f f f f 5 f 6 7 f End behavior: even odd f f f even odd f Sketch the end behavior for the following. f 5 f 4 5 f 5 f f 6 4 f 7 4 5
PROPERTIES OF POLYNOMIALS The -values that make a function = zero are called zeros of the function. Turning point--- a point where the graph goes from increasing to decreasing or from decreasing to increasing. Ma number of possible -intercepts --- the degree of the polynomial. Ma number of possible turning points --- (degree -) of the polynomial ) Label the zeros and give the locations of the turning points. Determine if the degree is odd or even and if the function has a negative leading coefficient. Sketch a graph of the polynomial function with the given information. ) Zeros =,5. even degree, positive leading coefficient. ) Zeros =-0,0,0. odd degree, negative leading coefficient.
ZEROS OF A POLYNOMIAL Find the zeros of a function: (even though imaginary numbers do not show up on a graph they can be considered zeros) P( ) 9 6 P( ) 9 4 Find the polynomial of least degree given the following zeros. ) =0,,- ) = 0,i,-i ) =5/, 4) Find a polynomial of degree with zeros of -, -, and 4. P()=( )( )( ) 5) Find a polynomial of degree 4 with zeros of, -i, and i. 6) Find a polynomial of degree with zeros of,, and (multiplicity of ) 4
DIVIDING POLYNOMIALS Section. Steps: ) ) ) 4) 909 q 4q q 4 q 4 q 4 q 4 q 4q 6 q 7q5 q 4 9 Synthetic Division Synthetic division can only be used when dividing with the form of r. q 7q5 q 4 9 i 5
Remainder Theorem P ( ) 5 6 Divide P ( ) 5 6 by P ( Find ) P ( ) 5 6 Divide P ( ) 5 6 by 4 Find P( 4) Remainder Theorem: If a polynomial P() is divided by -k, then the remainder is equal to P(k). If a polynomial P() is divided by -4, then the remainder is equal to P(4). If a polynomial P() is divided by +5, then the remainder is equal to P(-5). Find P (0.5) by substituting: P ( ) 5 6 by using synthetic division: 5 6 6
REAL ZEROS AND POLYNOMIAL INEQUALITIES Section.4 Finding Zeros of a polynomial. Determine if is a zero of P ( ) 5 6, then use synthetic division to find the other zeros. P ( ) 8 5 find the zeros? b b 4ac a P 4 ( ) 4 8 0, +i is a zero, find the otherzeros? 7
Solving an equation with a polynomial. 4 4 0 0 0 Hint: look for a repeater. Factoring. Find a polynomial of lowest degree with zeros of 7(multiplicity of ), and Use your zeros from the previous page to factor the following. Remember- use only rational zeros. Factor P ( ) 5 6 Factor P ( ) 8 5 4 Factor P ( ) 7 8 Finding Possible rational zeros. Multiply 5 + BLAH + Rational zeros are P ( ) 5 8 What are the possible rational zeros? Last number. Factors of last number First number. Factors of first number Possible Rational Zeros = 8
All together. Try: 5 8 0 ) list the possible zeros. ) Use synthetic division to find the rest of the zeros. ) Factor the polynomial. P ( ) Find the zeros (some may have a multiplicity) and find the factored form. P ( ) 4 4 4 49 64 40 P ( ) 4 4 4 P ( ) 5 75 9
DESCARTES'S RULE OF SIGNS Section C- Given a polynomial P() with real coefficients and nonzero constant term. ) Positive Zeros. The number of positive zeros of P() is the number of variations in the sign in P() or less by an even number. ) Negative Zeros. The number of negative zeros of P() is the number of variations in the sign in P(-) or less by an even number. Find the number of variations in sign in P(). 4 4 P ( ) 4 4 49 64 40 P ( ) 4 4 Find the number of real zeros and the number of imaginary zeros for each polynomial. 4 4 P ( ) 4 4 49 64 40 P ( ) 4 4 + i + i Graphs of polynomials. Use the leading term, end behavior, and multiplicity to sketch a rough graph of the polynomial. P() = ( + 4) () ( ) ( 7) P() = ( + 4) () ( ) ( 7) 0
Solving quadratic inequalities. INEQUALITIES AND POLYNOMIALS. 6 8 0 5 6 7 4 Section. ) Find the real zeros. ) set to zero. ) set to zero. ) Find the real zeros. ) Find the real approimate zeros.
WARM UP Graph the following on the same graph below and label the, y-intercepts. ) = ) =-5 ) y=4 4) y=- Draw a horizontal asymptote. What type of equation would describe this line? Draw a vertical asymptote. What type of equation would describe this line?
GRAPHING RATIONAL FUNCTIONS Section.5. factor first and reduce if possible f 8 9 VA =? Then find: a) Vertical asymptotes BOTTOM BAD set denominator=0 and solve. HA y=? b) Horizontal asymptotes Look at page 4
Find the VA. 4
FIND THE HORIZONTAL (the functions limit as approaches, what y is approaching) f() = atop degree + b bottom degree + Eamples of the three types of horizontal or oblique asymptotes: ) If top degree = bottom degree, then the horizontal asymptote is: y = a b f 6 lim f() = a b lim f() = a b ) If top degree is smaller than bottom degree, then the horizontal asymptote is: y=0, the -ais 4 f ( ) lim f() = 0 lim f() = 0 ) If top degree is larger than bottom degree, then the function does not have a horizontal asymptote. We have to divide to find the slant asymptote. f ( ) lim f() = or f() = or lim 0 y = + 5
Find the Horizontal or Slant Asymptote for each and draw it. 6
f 5 4 5 4 Canceled factors create holes---------------- X= Domain: A) FIND THE X-INTERCEPTS numerator 0 (, 0 ) X B) FIND THE Y-INTERCEPTS 0, f (0) ( 0, ) Y C) FIND THE VERTICAL ASYMPOTES D) HORIZONTAL/OBLIQUE ASYMPTOTE 7
4 f ( ) Domain: A) FIND THE X-INTERCEPTS numerator 0 (, 0 ) X B) FIND THE Y-INTERCEPTS 0, f (0) ( 0, ) Y C) FIND THE VERTICAL ASYMPOTES D) HORIZONTAL/OBLIQUE ASYMPTOTE 8
f ( ) Domain: A) FIND THE X-INTERCEPTS numerator 0 (, 0 ) X B) FIND THE Y-INTERCEPTS 0, f (0) ( 0, ) Y C) FIND THE VERTICAL ASYMPOTES D) HORIZONTAL/OBLIQUE ASYMPTOTE 9
Top = 0 Bottom = 0 5 0 Solving rational equations and inequalities. 5 0 5 0 6 5 5 4 6 5 0 5 4 6 5 0 5 4 6 5 0 5 4 4 4 4 0
VARIATIONS Section.6 DIRECT VARIATIONS Y=kX The k is called the constant of variation. "Y varies directly as " INVERSE VARIATIONS Y = k X "Y varies inversely as " JOINT/ COMBINED VARIATIONS Y varies jointly as and z Y = kz Y varies directly as and inversely as c Y = k c ) Write the variation ) Use the data to find k )Use the variation to answer the question. E/ Jimmy's income, I, varies directly as the number of apples, n, he sells. )Variation equation: If he earned $6 from the sale of 00 apples during one week, how many apples did he sell during another week if he earned $45? ) Find k: Variation equation: ) Answer the question using the variation: Try: a) Your time to school, T, varies inversely as you speed, s. If you needed 5 minutes to travel to school at a rate of 45 mph, then how much time did you need if you were traveling at a rate of 60mph? b) The weight gain of a Baboon, W, is varies jointly as the number of bananas, b, and the number of ants, a, he eats. If a baboon gained 5 pounds after eating 0 bananas and 50 ants, then how much would he gain after eating bananas and 00 ants? c) The bmal, b, varies directly as the tac, d, and inversely as the gar, g. If the bmal is 40 when the tac is 0 and the gar is 5, then what is the bmal when the tac is 5 and the gar is?