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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.
(3x4 + 2x3 − x) − (−x4 + x2 + x −1) =

Multiplying Polynomials

Two polynomials are multiplied by using Properties of
Real Numbers and Laws of Exponents .

Example: (4x5 ) ∙ (3x2 ) =

Example: Find the product.

(2x4 − 3x2 +1)(4x3 − x) =

Use FOIL when multiplying two binomials.
(FOIL – First, Outer, Inner, Last)

( y − 3)(2y + 5) =

Special Products and Factoring

Difference of Two Squares:

(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)(x2 + xy + y2 ) =

Sum of Two Cubes:

(x + y)(x2 − xy + y2 ) =

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) =

64x2 − 81=

(b) (7x + 5)2 =

4x2 + 28x + 49 =

(c) (2 − x)(4 + 2x + x2 ) =

8c3 + 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:
8x5 y3 + 6xy9 =

5x2 (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.
2x3 − 5x2 − 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

x2 + 9

y2 −10

Factoring a Second-Degree Trinomial

FOIL “in reverse” can be used for factoring the
trinomials over the integers.
(ax + b)(cx + d ) = ac ∙ x2 + (ad + bc) ∙ x + bd

Example: Factor the trinomials.

2x2 + 5x − 3 =

6x2 −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 =

x6 + 7x3 − 8 =

5(3 − 4x)2 − 8(3 − 4x)(5x −1) =

b6 − 27 =

x6 − y6 =

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
x3 + x2 −10x − 6

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