# Polynomials in One Variable

An expression of the form

where k is a whole number and is a constant,
is

called a monomial. A sum of monomials forms a

polynomial where each monomial is called a term.

Polynomial in Standard Form:

where are coefficients, and n ≥ 0 is an

integer.

If , then

n – degree,

– leading coefficient,

– leading term.

Polynomial? Yes/No Coefficients Degree

**Adding and Subtracting Polynomials **

Polynomials are added and subtracted by combining

like terms. The like terms are the monomials which

may differ only by the coefficients.

Example. Perform the indicated operation .

(3x^{4} + 2x^{3} − x) − (−x^{4} + x^{2} + x −1) =

Two polynomials are multiplied by using Properties of

Real Numbers and Laws of Exponents .

Example: (4x^{5} ) ∙ (3x^{2} ) =

Example: Find the product.

(2x^{4} − 3x^{2} +1)(4x^{3} − x) =

Use FOIL when multiplying two binomials.

(FOIL – First, Outer, Inner, Last)

( y − 3)(2y + 5) =

**Special Products and Factoring **

(x − y)(x + y) =

Squares of Binomials, or Perfect Squares:

(x + y)^{2} =

(x − y)^{2} =

Cubes of Binomials , or Perfect Cubes:

(x + y)^{3} =

(x − y)^{3} =

Difference of Two Cubes:

(x − y)(x^{2} + xy + y^{2} ) =

Sum of Two Cubes:

(x + y)(x^{2} − xy + y^{2} ) =

Factoring is a process of finding polynomials whose

product is equal to a given polynomial.

Example: Expand or factor by using the special product

formulas :

(a) (x + 3)(3− x) =

64x^{2} − 81=

(b) (7x + 5)2 =

4x^{2} + 28x + 49 =

(c) (2 − x)(4 + 2x + x^{2} ) =

8c^{3} + 27 =

(d) (2x − 3)^{3} =

**Factoring out the Common Factor :**

The CF of a polynomial is formed as a product of the

factors (numbers, variables , and/or expressions)

common to all terms, each raised to the smallest power

that appears on that factor in the polynomial.

Remember when factoring out the CF, we use the

Distributive property

ab + ac = a(b + c)

that is, we divide each term by the CF.

Example:

8x^{5} y^{3} + 6xy^{9} =

5x^{2} (x − 2)^{3} + x(x − 2)^{2} =

**Factoring by Grouping**

This method is used when the terms can be collected in

two or more groups such that there is a common factor

in all groups.

Example: Factor by grouping.

2x^{3} − 5x^{2} − 8x + 20 =

**Prime (Irreducible) Polynomials
**A polynomial is called prime or irreducible over a

specified set of numbers if it cannot be written as a

product of two other polynomials whose coefficients

are from the specified set.

A polynomial is considered to be factored completely

over the particular set of numbers if it is written as a

product of prime over that set polynomials.

Example: Determine which of the polynomials below

is/are prime over the real numbers

x^{2} + 9

y^{2} −10

**Factoring a Second-Degree Trinomial**

FOIL “in reverse” can be used for factoring the

trinomials over the integers.

(ax + b)(cx + d ) = ac ∙ x^{2} + (ad + bc) ∙ x + bd

Example: Factor the trinomials.

2x^{2} + 5x − 3 =

6x^{2} −17x +12 =

**Factoring by substitution**

Example: 16(x +1)^{2} + 8(x +1) +1=

Example: Factor completely over the integers by any

method.

(x −1)^{3} − 64 =

x^{6} + 7x^{3} − 8 =

5(3 − 4x)^{2} − 8(3 − 4x)(5x −1) =

b^{6} − 27 =

x^{6} − y^{6} =

**Polynomial Division**

**Long Division **

426 =

Check: Dividend = (Quotient)(Divisor) + Remainder

Dividing by a monomial:

Dividing two polynomials with more than one term:

(1) Write terms in both polynomials in descending

order according to degree.

(2) Insert missing terms in both polynomials with a 0

coefficient.

(3) Use Long Division algorithm. The remainder is a

polynomial whose degree is less than the degree of

the divisor.

Example: Perform the division.

** Synthetic Division
**Synthetic division is used when a polynomial is divided

by a first-degree binomial of the form x − k .

←Coefficients of Dividend

Diagonal pattern: Multiply by k

Vertical pattern: Add terms

Example: Use synthetic division to find the quotient

and remainder.

Example: Verify that x − 3 is a factor of

x^{3} + x^{2} −10x − 6

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