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

\(\text{a)} \ 3x\cdot(-5x^3+2x+2)=-15x^4+6x^2+6x\)

\(\begin{align}\text{b)} \ (x^2-1)\cdot(4x+3)&=x^2(4x-3)-1(4x-3) \\ &=4x^3-3x^2-4x+3\end{align}\)

\(\begin{align}\text{c)} \ (-4x^2+2x+5)\cdot(2x-5)&=-4x^2(2x-5)+2x(2x-5)+5(2x-5) \\ &=-8x^3+20x^2+4x^2-10x+10x-25 \\ &=-8x^3+24x^2-15\end{align}\)