Question

polynomials introduction
they are functions that fit a specific description.

• they all graph as either a line or a smooth curve.
• they are all functions (each x - value is used only once)
• they usually have two variables (such as x and y) but a special case exists where they have only one variable (such as y = 3)
• the independent variable may be raised to a power but the powers must be whole numbers.
• there may be many terms in the polynomial.
• each term will have a coefficient - if the coefficient isnt written in then it is 1.
• there cannot be variables in the denominator
• the variable cannot be inside a radical sign

vocabulary used for polynomials
term:
coefficient:
constant:

circle the equations that are polynomials. if the equation is not a polynomial, draw an arrow to the part of the equation that disqualifies it.
$y = 6x^{2}+4x - 3$
$y = 7$
$y=\frac{2x + 4}{3x^{2}+5}$
$y = 5x^{8}+6x^{6}+4x^{3}$
$y = 2sqrt{x}+4x$
$y = x^{2}$
$y = 4xz+5x^{2}z - 8z$
$y = 20x - 5$
$y = 7x^{3}+3x^{2}-8$

Explanation:

Step1: Check polynomial rules

A polynomial has variables raised to non - negative integer powers, no variables in denominators, no variables inside radical signs.
For \(y = 6x^{2}+4x - 3\):
Variables \(x\) have non - negative integer powers (\(x^{2}\) and \(x=x^{1}\)), no variables in denominator or radical.
For \(y = 7\) (can be written as \(y=7x^{0}\)):
Variable \(x\) has power \(0\) (non - negative integer).
For \(y=\frac{2x + 4}{3x^{2}+5}\):
There is a variable \(x\) in the denominator (\(3x^{2}+5\) is in the denominator of the fraction).
For \(y = 5x^{8}+6x^{6}+4x^{3}\):
Variables \(x\) have non - negative integer powers (\(x^{8},x^{6},x^{3}\)).
For \(y = 2\sqrt{x}+4x\) (\(y = 2x^{\frac{1}{2}}+4x\)):
Variable \(x\) has a non - integer power (\(\frac{1}{2}\)).
For \(y=x^{2}\):
Variable \(x\) has non - negative integer power (\(x^{2}\)).
For \(y = 4xz+5x^{2}z - 8z\):
Variables \(x\) and \(z\) have non - negative integer powers (\(x^{1},z^{1},x^{2},z^{1}\)).
For \(y = 20x-5\):
Variable \(x\) has non - negative integer power (\(x=x^{1}\)).
For \(y = 7x^{3}+3x^{2}-8\):
Variables \(x\) have non - negative integer powers (\(x^{3},x^{2}\)).

Answer:

• Polynomials: \(y = 6x^{2}+4x - 3\), \(y = 7\), \(y = 5x^{8}+6x^{6}+4x^{3}\), \(y=x^{2}\), \(y = 4xz+5x^{2}z - 8z\), \(y = 20x-5\), \(y = 7x^{3}+3x^{2}-8\)
• Not polynomials:
• For \(y=\frac{2x + 4}{3x^{2}+5}\), the non - polynomial part is \(\frac{2x + 4}{3x^{2}+5}\) (variable in denominator).
• For \(y = 2\sqrt{x}+4x\), the non - polynomial part is \(2\sqrt{x}\) (variable inside radical).