Understanding Polynomials

Intro to Polynomials

Classify and Simplify Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients. Variables are also sometimes called indeterminates. We can perform arithmetic operations such as addition, subtraction, multiplication, and also positive integer exponents for polynomial expressions but not division by variable.

A math expression containing two or more algebraic terms

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Ex: 4x⁵​ + 2x⁴ - 3x³ + 8x² + 7x - 10

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​Polynomials Definition

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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.

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

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

POLYNOMIAL

• Monomial

• Binomial

• Trinomial

• Multinomial

What it IS and what it is NOT:

DEGREE OF POLYNOMIAL

The degree of a polynomial in one variable is determined by the highest exponent of the variable in the expression.

Leading coefficient is the numerical coefficient of the leading term.

Constant term is the term that do not contains variable.

Standard Form

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

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.

Adding and Subtracting Polynomials

• Step 1: Distribute the "1s" to eliminate the parentheses.

• Step 2: Combine like terms

Ex 1 (-7.1x²-180x+5800)+(21x²-140x+1900)

• Since the number in front of both sets of parentheses is 1, this eliminates the parentheses

• -7.1x²-180x+5800+21x²-140x+1900

• Combine like terms

• 13.9x²-320x+7700

Ex 2 (x³-3x²+5x)-(7x³+5x²-12)

• Step 1: Notice there is a -1 in front of the 2nd set of parentheses. This changes the signs of the second polynomial.

• x³-3x²+5x-7x³-5x²+12

• Step 2: Combine Like Terms

• -6x³-8x²+5x+12