Intro to Polynomials

Intro to Polynomials

Classify and Simplify Polynomials

What is a Polynomial?

A polynomial is an expression involving a sum of powers in one or more variables multiplied by coefficients, where the powers must be whole.

What it IS and what it is NOT:

 $\sqrt{x}$  is a non-whole number exponent.
 $x^{-2}$  and  $x^{-1}$  are non-whole number exponents. 

Multiple Select

Which algebraic expressions are polynomials?

• $\pi x-\sqrt{3}+5y$

• $x^2y^2-4x^3+12y$

• $\frac{4}{x}-x^2$

• $\sqrt{x}-16$

• $3.9x^3-4.1x^2+7.3$

Classifying Polynomials

A polynomial can be classified by the number of terms it has. Each term of a polynomial is separated by addition or subtraction.

Classifying Polynomials

The degree of a polynomial is the highest exponent or the sum of exponents on any variable term.

Standard Form

Polynomials in standard form have the highest degree variable term followed by descending powers of the variable terms.

How can we classify this polynomial?

More Standard Form

For multivariable terms, there can be multiple standard forms for the polynomial.

The top example is in descending powers of x, while the bottom example is in descending powers of y. Both examples are fourth degree polynomials in standard form.

Multiple Choice

Determine the standard form of the given polynomial:

 $\frac{x}{4}-2x^5+\frac{x^3}{2}+1$  

•  $\frac{x^3}{2}-2x^5+\frac{x}{4}+1$  

•  $-2x^5+\frac{x^3}{2}+\frac{x}{4}+1$  

•  $-2x^5+\frac{x}{4}+\frac{x^3}{2}+1$  

•  $1-2x^5+\frac{x^3}{2}+\frac{x}{4}$  

Simplifying Polynomials

Simplify the polynomial by combining like terms, and write the polynomial in standard form. Classify the polynomial by number of terms and degree.

Multiple Choice

Which polynomial correctly combines like terms and puts the given polynomial in standard form?

 $\frac{1}{3}x+8x^4-\frac{2}{3}x^2-x$  

•  $8x^4-\frac{1}{3}x^2-x$  

•  $8x^4+\frac{2}{3}x^2+\frac{2}{3}x$  

•  $8x^4-\frac{2}{3}x^2-\frac{2}{3}x$  

•  $8x^4-\frac{2}{3}x^2-x$