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Polynomials in One Variable
Polynomial in Standard Form:
where are coefficients, and n ≥ 0 is an
If , then
n – degree,
– leading coefficient,
– leading term.
Polynomial? Yes/No Coefficients Degree
Adding and Subtracting Polynomials
Example. Perform the indicated operation.
(3x4 + 2x3 − x) − (−x4 + x2 + x −1) =
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.
(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
ab + ac = a(b + c)
that is, we divide each term by the CF.
8x5 y3 + 6xy9 =
5x2 (x − 2)3 + x(x − 2)2 =
Factoring by Grouping
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
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
(x −1)3 − 64 =
x6 + 7x3 − 8 =
5(3 − 4x)2 − 8(3 − 4x)(5x −1) =
b6 − 27 =
x6 − y6 =
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
(3) Use Long Division algorithm. The remainder is a
polynomial whose degree is less than the degree of
Example: Perform the 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
Example: Verify that x − 3 is a factor of
x3 + x2 −10x − 6