Introduction to Polynomials
Variable expressions: 3x + Sy - 4; x2 - 3x + 1; Sxy - x2y2 + 24; 8; x, etc.
A polynomial is a variable expression in which the terms are monomials.
Apolynomial of one term is a monomial. 3x2 , X ) XH~
A polynomial of two terms is a binomial. 3x2 + 2
A polynomial of three terms is a trinomial. 3x2 + 2x - 1
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IThe degree of a polynomial is the greatest of iho-dog£ges-of any of its terms, it is
9 always a positive integer.
The leading coefficient of a polynomial function is the coefficient of the term with the
greatest exponent on a variable.
The constant term is the term without a variable.
Example:
a) 10x7 - 2x6 + xs - 3x4 - x3 + 2x2 - Wx + -
Degree 7; Leading coefficient 10;
Constant 1/4
b) 4x - 3x2 - 7x5
Degree 5; Leading coefficient -7
Constant 0
The terms of a polynomial in one variable are usually arranged so that the exponents on
the variable decrease from left to right. This is called descending order.
Example:
4x - 3x2 - 7xs = -7x5 - 3x2 + 4x
Notation of a polynomial: P(x); Q{x),M{x)lF{x), etc.
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Exercise 1:
1. Find the leading coefficient, the constant term, and the degree of the polynomial:
P(x) = Sx6 - 4x5 - 3x2 + 7
NOT Polynomials: s(x) =
*+1; v(x) = yJx-3. q(x) = x<&- 11
lu (V) •> V-l V —C
Polynomials: P(x) =x9 - 3x5 +1; qr(*) =I*i^*±i *i y2_ ^y _^ |
Exercise 2:
1.
Indicate whether P(x) = -jc2 +5x +4 is a polynomial. If it is a polynomial then;
a. Identify the leading coefficient.- /
b. Identify the constant term. t-f.
c. State the degree.
2.
O
Indicate whether P(x) = x2 -5xA -x6 is a polynomial. If it is a polynomial then:
a. Identify the leading coefficient-\
b. Identify the constant term. 0
c. State the degree. ft Vp
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,
.
^ ^(V^N- —V * ;nV'
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Polynomial with degree 1(first degree) is called LINEAR POLYNOMIAL. &*'• r^ ~^C "3
Polynomial with degree 2 (second degree) is called QUADRATIC POLYNOMIAL.
Polynomial with degree 3 (third degree) is called CUBIC POLYNOMIAL.
Polynomial with degree 4 and above are called "4th degree "POLYNOMIAL; "5th degree"
POLYNOMIAL etc.
OPERATION WITH POLYNOMIALS
1. Evaluating polynomial expressions
To evaluate a polynomial function, replace the variable by its value and simplify.
Example: Given /?(x) = -x3 + 3x2-8x +5, evaluate R(-\).
2>
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X ——l
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- -^0 +3-v -<?CrU i f
•
Exercise 3:
1. Given P(x) =-4x2 -6x +5, evaluate P(-3). P£-£) - - H(~S)2' (o (- 3) +~5
~ -Htf) -tig ±5
2. Given/(x) =x5-2x' +5x, evaluate/(l).
=
-13
to5-w +«?co
2. Adding and subtracting variable expressions
Polynomials can be added by combining like terms. Either a vertical or a horizontal
format can be used.
Like terms are the terms with the same variable and the same exponent
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Vertical format
Example: Add (2x3 + Sx2 - 7x +1) + (-x3 - Sx2 + 3x - 6) using a vertical format.
dx^-t 5XZ~1K+ (
xVox7-- 4x -£
X*> -4x -5
The additive inverse of the polynomial (x2 + 3x - 5) is -(x2 + 3x - 5) = -x2 - 3x + 5
Example: Subtract (3x4 - 2x2 + Sx - 3) - (x3 + 5x2 + 3x - 4) using vertical format.
-X -5X2- -V< h-1,
Exercise 4:
1. Add the following. Use a vertical format.
(3x2+3x +2) +(4;c2+7;c-6)
2. Subtract the following. Use a vertical format.
(2x2+9x +4)-(4*2 -5x +10)
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Horizontal format
Example:
1. Add (2x3 + 5x2 - 7* + 1) + (-x3 - Sx2 + 3x - 6) using a horizontal format.
———
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2. Subtract (3x4 - 2x2 + 5x - 3) - (x3 + 5x2 + 3x - 4) using horizontal format.
- 2>x4-K5-7xx+2.x-+ I
Exercise 5: Add the following. Use a horizontal format.
{-9x2 +2* +5) +(2jc2 +4x-12)
- -n^vvpY'i
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2.
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-C--T "f '3 -i -ti
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y
-Z
0
J Cx),
xMx
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(-3)3- H(-2) - -Z f f * 0
(-I)3 -HC-I) - -I 4-4=3
(Of-H-(O) =0-0-0
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(tf-nO) = / - ¥ * -3
%>(*W 3^ ^Z5
SCt) =^.735T|
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-5(3) =. >735Tt3i
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= \S
tx- <?£> h ^
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P(x)- 2>x -o.v-e /
P-(x)^^¥k -~<o~r~p
St» = ?C<) -f R(xj
v
f / ./. 3x5- 5*
-s
3
fcx III 33l£
(AxVaxSi, x+ff) T(x%xi5X-2>x^xW?
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(&x% 4x^ax-/) ^(^fx\3x L-+ i) -^a3-3
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2.
-4x +3x .= -x
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t?x 74/32 &
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TO;-- ^i^AxJh^+c
inc(
?6g - 9. -2- k-3. %a- 2 -r- c
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J, = \(o - IG -^ -t- C_
3 ^ -^+c
c
= 7.
2