Introduction to Polynomials: Key Facts

Introduction to Polynomials: Key Facts

What is a polynomial?

• A polynomial is an expression of the form a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 + . . . + a ₁(x) ¹+ a₀

• What does that look like?

• 5x⁴ + x³ - 6x² + 2x + 8

• Remember that a term is a constant, a variable, a variable to a degree, or the product of a constant and variable. Terms are joined by addition or subtraction.

Standard Form of a Polynomial

• A polynomial is written with the largest degree first

• It is written in descending order, each next term has a degree less than the term before it.

• The constant always comes last, since it is always a number * x⁰

• 3x² + x + 4x³ + 10 in standard form is 4x³ + 3x² + x + 10

Standard Form of a Polynomial

• When there is more than 1 variable name in the polynomial, the terms with the variable that appears first in the alphabet will appear in descending order

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Parts of a Polynomial

• COEFFICIENT is the number in front of the variable. It must be a REAL NUMBER.

• The LEADING COEFFICENT is the coefficient of the first term

• The VARIABLE is a letter that represents an unknown value

• The EXPONENT is the power of the variable. It must be an INTEGER.

Degree of a Polynomial

• The degree of a polynomial is the greatest sum of the degrees of the variables in each term

• When adding up the exponents of a term, remember to count a variable with no variable showing as degree 1

• Degree of a variable, or constant, is the exponent of that variable

• The constant term is multiplied by an x⁰, which is 1

Degree of a Polynomial

• The degree of the term is the sum of the degrees of the variables (or constants) in that term

• In the example, the term with the greatest sum of exponents is the first term, = 9

• The degree of the polynomial is 9, the greatest sum of the exponents in a term.

Naming Polynomials by Degree

• Constant = Degree 0

• Linear = Degree 1

• Quadratic = Degree 2

• Cubic = Degree 3

• Quartic = Degree 4

• Quintic = Degree 5

• For anything over degree 3, it is sometimes just named a polynomial of degree _______

Naming a Polynomial by Number of Terms

• One term: monomial

• Two terms: binomial

• Three terms; trinomial

• Four terms: quadrinomial

• Usually, a polynomial of more than 3 terms is just called a polynomial.

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Multiple Choice

Write the polynomial so the powers of x are in descending order.

9x² + 5x + 27 + 2x³

• 27 + 5x + 9x² + 2x³

• 9x² + 5x + 2x³ + 27

• 9x² + 5x + 27 + 2x³

• 2x³ + 9x² + 5x + 27

Multiple Choice

Which one is written in Standard form

• 4 + 5b³

• 4n³ - n + 6n²

• 2a² + 9

• 2h⁵ - 5h³ - 8 + 3h

Multiple Choice

What is the degree of this term?
-7xyz²

• 3
• 4
• -7
• 7

Multiple Choice

What is the correct name of the following:

3xy + 4x + 5y + 6

• monomial

• binomial

• trinomial

• polynomial

Multiple Choice

What type of polynomial is this and what is its degree?

5b³ - 3 + 6b

• 3rd degree binomial

• 3rd degree monomial

• 3rd degree trinomial

• 1st degree polynomial

Multiple Choice

Which one is NOT a monomial

• 3x - 4

• 12z

• 14v⁸w

• -12x²

Multiple Choice

 $\frac{12}{g^3}$  is a polynomial. 

• True

• False

Multiple Choice

How many terms does the following polynomial have?

8x⁵ - 5x⁴ + 3x³ - 7x² + 5x - 19

• 6

• 5

• 15

• 14

Fill in the Blanks

 $Given\ :\ f\left(x\right)=3x^4-2x^2+9x\ +4$  

What is its leading coefficient?

Fill in the Blanks

 $Given\ :\ f\left(x\right)=5x^2\ +6x\ -\ 2$  

What is its leading coefficient?