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1. Definition of polynomial
Polynomial is a function of the form
| \(p(x)=a_{n}x^n+a_{n-1}x^{n-1}+…a_{1}x^1+a_{0}x^0\) |
In the above expression x is the independent variable. \(a_{n}, a_{n-1}, …, a_{1}\) and \(a_{0}\) are the coefficients where \(a_{n}\) is known as the leading coefficient.
The degree of the polynomial p(x) above is given by n where n is a non-negative integer. Note that \(1\over x\) is not a polynomial because the power is -1.
Some polynomials have special names. Polynomial with degree n=0,
is known as a constant function. Polynomial with degree n=1,
is known as a linear function. Some other names are shown below.
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Degree, n
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Name
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0
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Constant function
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1
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Linear function
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2
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Quadratic function
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3
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Cubic function
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4
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Quartic function
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5
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Quintic function
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When a polynomial is written starting with the term of the highest degree, the terms are said to be in a descending order. When a polynomial is written starting with the term of the lowest degree, the terms are said to be in an ascending order.
For instance,
is a fourth degree (n=4) polynomial in descending order. The leading coefficient is \(a_{n}=3\).
Also,
is a third degree (
n=3) polynomial in ascending order. The leading coefficient is \(a_{n}=-1\).
2. Addition and subtraction
To add or subtract polynomials, you add or subtract the coefficients of the same power. \
Example
Given \(p(x)=x^3+2x+7 \text{ and } q(x)=5+3x^2-x^3\),
\(p(x)+q(x)=x^3+2x+7 +5+3x^2-x^3\)
\(=1+(-1)x^3+(0+3)x^2+(2+0)^x+(7+5)\)
\(=3x^2+2x+12\)
\(p(x)-q(x)=(x^3+2x+7) -(5+3x^2-x^3)\)
\(=(1-(-1)x^3+(0-3)x^2+(2-0)x+(7-5)\)
\(=2x^3-3x^2+2x+2\)
3. Multiplication
Multiplying with a constant
To multiply a polynomial with a constant, you simply multiply each term in the polynomial with that constant.
Example
Given \(p(x)=3x^4-x^3+2x+7\), find 5p(x).
\(5p(x)=5×(3x^4-x^3+2x+7)\)
\(=15x^4-5x^3+10x+35\)
Example
Given \(p(x)=x^2-2x+3\) and \(q(x)=x^2+x+1\), find p(x)+3q(x).
\(p(x)+3q(x)=x^2-2x+3+3(x^2+x+1)\)
\(=x^2-2x+3+3x^2+3x+3\)
\(=4x^2+x+6\)
Multiplying two polynomials
To multiply two polynomials, you multiply each term in one polynomial with all the terms in the other polynomial.
Example
Given \(p(x)=x^3+5 \text{ and } q(x)=3x^2-x\), find \(p(x)×q(x)\).
\(p(x)×q(x)=x^3+5(3x^2-x)\)
\(=x^3(3x^2-x)+5(3x^2-x)\)
\(=3x^5-x^4+15x^2-5x\)
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