Polynomials#
poly = “many”, nomial = “term”
Polynomial terms involve only natural number exponents. Negative exponents, fractional exponents and variable roots are exluded.
For example
a) \(P(x)=3x^2\) is monomial and so are constants
b) \(P(x)=3x^2-x\) is binomial
c) \(P(x)=3x^2-x+9\) is trinomial
d) \(P(x)=3x^{10}+6x^5-x^4+12x^2-4\) is polynomial, as are all the above
In general \(P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_2x^n+a_1x+a_0\ (a_n\in\mathbb{R}{,}\ n\in\mathbb{N})\) in standard form of polynomials, where the terms with the highest degree are first and \(a_0\) has an exponent of zero and is called a constant.
The degree of a polynomial is the highest of the degrees of the polynomial’s monomials (highest exponent of the variables) with non-zero coefficients. If a polynomial has more than one variable, the degree of a polynomial is the largest sum of the exponents in a single term.
For example
a) \(P(x{,}y)=4xy^2+2xy-x+4y-1\) has the degree of 3
b) \(Q(a{,}b{,}c)=-6ab+a^2c+3abc-2ab^2=-2ab^2+a^2c+3abc-6ab\) has the degree of 3
If a polynomial needs to be simplified, remember that only like terms can be combined.
Examples#
addition
\(\begin{align}(-5x^3+2x^2-5)+(7x^3+3x^2+5)&=7x^3-5x^3+3x^2+2x^2+5-5 \\ &=2x^3+5x^2\end{align}\)
subtraction
\(\begin{align}(-5x^3+2x^2-5)-(7x^3+3x^2+5)&=-5x^3+2x^2-5-7x^3-3x^2-5 \\ &=-5x^3-7x^3+2x^2-3x^2-5-5 \\ &=-12x^3-x^2-10\end{align}\)
multiplication
\(\text{a)} \ 3x\cdot(-5x^3+2x+2)=-15x^4+6x^2+6x\)
\(\begin{align}\text{b)} \ (x^2-1)\cdot(4x+3)&=x^2(4x-3)-1(4x-3) \\ &=4x^3-3x^2-4x+3\end{align}\)
\(\begin{align}\text{c)} \ (-4x^2+2x+5)\cdot(2x-5)&=-4x^2(2x-5)+2x(2x-5)+5(2x-5) \\ &=-8x^3+20x^2+4x^2-10x+10x-25 \\ &=-8x^3+24x^2-15\end{align}\)
Division of polynomials#
Division is done by factoring the numerator and/or the denominator in order to find something to simplify from both sides of the quotient. An algorithm called division angel and grouping are useful and will be covered in the next chapter called Factoring. If the division isn’t even, what will be left is a remainder.
For eaxmple
\(\frac{(3x^3-7x^2+5x-6)}{(x-2)}=\frac{(x-2)(3x^2-x+3)}{(x-2)}=3x^2-x+3\)