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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.
(3x^{4} + 2x^{3} − x) − (−x^{4} + x^{2} + x −1) =
Multiplying Polynomials
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
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)(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 SecondDegree 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 firstdegree 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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