a monomial or sum of monomials · advalg11.1introductiontopolynomials.notebook 2 may 11, 2018 apr...
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AdvAlg11.1IntroductionToPolynomials.notebook
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Polynomial:
Polynomial in x is an expression of the form
anxn + an 1xn 1 + an 2 xn 2 + ……. a1x1 + a0
where n is a positive integer and an ≠ 0
A monomial or sum of monomials
Example: 6x3 + 2x ─ 8x4 + 4
Standard Form: Terms written in descending order by exponent
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.
constantterm degree 0
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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.
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.
All exponents of the polynomial are integers greater than 0
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(5x3 6)(5x3 6)
25x6 30x3 30x3 + 36
25x6 60x3 + 36
6
25
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A first degree polynomial.
ax + b
A second degree polynomial.
ax2 + bx + c
A third degree polynomial.
ax3 + bx2 + cx + d
A fourth degree polynomial.
ax4 + bx3 + cx2 + dx + e
All nonzero constants are considered polynomials. The degree of a nonzero constant is zero. 3 = 3x0
The number zero is not considered a polynomial because all leading coefficeints must be nonzero. The degree of the number zero is undefined.
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P(1) = (1)5 4(1)4 + (1)2 5(1) + 50
= 1 4(1) + 1 5 + 50 = 1 4 +1 + 5 + 50
= 51
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21 22 23 24 25 26 275000 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
7934.372938.662720.98
2519.422332.802160.002000
A(x)=5000x6+2000x5+2000x4+2000x3+2000x2+2000x+2000
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Classifying Polynomials by Degree
Special Type Definition Example
Linear
Quadratic
Cubic
Quartic
Polynomials of the first degree
Polynomials of the second degree
Polynomials of the third degree
Polynomials of the fourth degree
mx + b
ax2 + bx +c
ax3 + bx2 +cx + d
ax4 + bx3 +cx2 + dx + e
not all degrees have to be included in the polynomial
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Expanding the Polynomial
1. ( 2x ─ 7 ) 2 2. ( 4x + 5 ) 3
Polynomial Functions
1. p(x) = x5 ─ 4x4 + x2 ─ 5x + 50
Find p ( 1 ) =
Find p ( 0 ) =
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Graph polynomial listed above using the following window:
5 ≤ x ≤ 5 and 60 ≤ y ≤ 60 scale of 1 scale of 10
Sketch of graph:
p(x) = x5 ─ 4x4 + x2 ─ 5x + 50
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Real Life Application1. 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 5th year?
2. Mark received $250 on his 16th birthday. On each birthday after his 16th, 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?
P ( 1 + r)n
$842.45
$1980.75
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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?
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