Polynomial: A monomial or sum of monomials Polynomial in x is an expression of the form a n x n + a n 1 x n 1 + a n 2 x n 2 +. a 1 x 1 + a 0 where n is a positive integer and a n 0 Example: 6x 3 + 2x 8x 4 + 4 Standard Form: Terms written in descending order by exponent constant term degree 0 Leading Coefficient: Coefficient of the term with the highest degree. Degree of the polynomial: Highest exponent of any of the terms after the polynomial has been simplified. Mar 1 12:54 PM 1
AdvAlg11.1IntroductionToPolynomials.notebook When the polynomial contains only one variable the degree of the polynomial is the largest exponent of the variable. The expressions of the polynomial that are being added and/or subtracted are the terms of the polynomial. All exponents of the polynomial are integers greater than 0 The terms must be written in descending order by exponent. The numbers multiplied by each variable expression. The number multiplied by the expression with the highest exponent. Multiplying polynomials or raising a polynomial to an exponent and then simplifying. Apr 22 9:51 AM 2
(5x 3 6)(5x 3 6) 25x 6 30x 3 30x 3 + 36 25x 6 60x 3 + 36 6 25 Apr 22 10:03 AM 3
A first degree polynomial. ax + b A second degree polynomial. ax 2 + bx + c A third degree polynomial. ax 3 + bx 2 + cx + d A fourth degree polynomial. ax 4 + bx 3 + cx 2 + dx + e All nonzero constants are considered polynomials. The degree of a nonzero constant is zero. 3 = 3x 0 The number zero is not considered a polynomial because all leading coefficeints must be nonzero. The degree of the number zero is undefined. Apr 22 10:07 AM 4
P( 1) = ( 1) 5 4( 1) 4 + ( 1) 2 5( 1) + 50 = 1 4(1) + 1 5 + 50 = 1 4 +1 + 5 + 50 = 51 Apr 22 10:18 AM 5
21 22 23 24 25 26 27 5000 5000(1.08) 6 2000 2000(1.08) 5 2000 2000(1.08) 4 2000 2000(1.08) 3 2000 2000(1.08) 2 2000 2000(1.08) 1 2000 7934.37 2938.66 2720.98 2519.42 2332.80 2160.00 2000 A(x)=5000x 6 +2000x 5 +2000x 4 +2000x 3 +2000x 2 +2000x+2000 Apr 22 10:41 AM 6
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Classifying Polynomials by Degree Special Type Definition Example not all degrees have to be included in the polynomial Linear Quadratic Cubic Polynomials of the first degree Polynomials of the second degree Polynomials of the third degree mx + b ax 2 + bx +c ax 3 + bx 2 +cx + d Quartic Polynomials of the fourth degree ax 4 + bx 3 +cx 2 + dx + e Mar 1 1:10 PM 24
Expanding the Polynomial 1. ( 2x 7 ) 2 2. ( 4x + 5 ) 3 Polynomial Functions 1. p(x) = x 5 4x 4 + x 2 5x + 50 Find p ( 1 ) = Find p ( 0 ) = Mar 1 1:23 PM 25
p(x) = x 5 4x 4 + x 2 5x + 50 Graph polynomial listed above using the following window: Sketch of graph: 5 x 5 and 60 y 60 scale of 1 scale of 10 Mar 1 1:26 PM 26
Real Life Application P ( 1 + r) n 1. Lori invested $150 at the beginning of each year, for 5 years. There were no additional deposits or withdrawals made. If Lori earned 3.9% interest, how much was in her account at the end of the 5 th year? $842.45 2. Mark received $250 on his 16 th birthday. On each birthday after his 16 th, the amount he received increased by $50. Mark invested the money in an account paying 7.2% interest and did not make any additional deposits or withdrawals. How much money did Mark have on the day he turned 20? $1980.75 Mar 1 1:31 PM 27
You invest $500 each Jan 1st year from age 14 through age 21. The money is left in the account until you retire at the end of the year you turn 65. If no additional deposits or withdrawals are made, and the interest earned is approximately 12%, how much money would you have in the account? What if you waited and started your deposits one year later? How much money would it cost you? May 4 7:05 AM 28