Polynomials
Common Mistakes
Polynomials
Definition
Common Mistakes
A polynomial is a single term or a sum
or difference of terms in which all
variables have whole-number exponents
and no variable appears in the
denominator.


Each term can be either a constant, a
variable, or a combination of
coefficients and variables.


The numerical part of the term is the
coefficient.

The highest power is the degree of the
polynomial.

Not a polynomial:
6 x 2 + 3 x −3
5
y2 + − 4 y + 3
y
Polynomial:
6 x 3 + 3x 2
y2 − 4 y + 3
Incorrect:
The coefficient of the term − 6 xy is 6.
The degree of the polynomial
3 x 3 − 6 x 5 − 4 x + 2 is 3.

Correct:
The coefficient of the term
The degree of the polynomial
3 x 3 − 6 x 5 − 4 x + 2 is 5.
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− 6 xyis
− .6
Polynomials
Types of Polynomials


Monomial-A constant, or the
product of a constant, and one or
more variables raised to a whole
number.
Example: − 6 x 2 y 3 z
Polynomial-Any finite sum (or
difference) of terms.
Example:
4 x 3 y 2 − 3 z + 9 x 2 y − 2 xz 3
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
Binomial-A polynomial consisting
of exactly two terms.
Example: 2 x − 7

Trinomial-A polynomial consisting
of exactly three terms.
Example:
x3 − x + 4
Polynomials-Adding/Subtracting
How to Add and Subtract
Polynomials


To add or subtract polynomials
combine like terms (group
together the same variable terms
with the same degrees).
When subtracting, if the
subtraction sign (or negative sign)
is outside of a parenthesis, you
must distribute the negative sign
to each of the terms inside the
parenthesis.
Common Mistakes

Addition
Simplify : (3 x 2 − 4 x + 5) + (−2 x 2 + 2 x − 6)

Incorrect:
x 4 − 2 x 2 − 1 or − x 6 − 1

Correct:
x2 − 2x −1

Subtraction
Simplify : (3 x 2 − 4 x + 5) − (−2 x 2 + 2 x − 6)

Incorrect:
3x 2 − 4 x + 5 + 2 x 2 + 2 x − 6

Correct:
3x 2 − 4 x + 5 + 2 x 2 − 2 x + 6
= 5 x 2 − 6 x + 11
Complete Manual: ..\Polynomial Review.docx
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Polynomials-Multiplying
How to Multiply
Polynomials


To multiply a monomial by a
monomial multiply the coefficients
together then multiply the
variables using the same rules that
apply as with exponents.
To multiply a monomial and a
polynomial distribute the
monomial across the polynomial.
Follow the same rules as with
multiplying monomials
Common Mistakes

Multiply monomials
Simplify : (−3 x 2 y 4 )(−2 x 3 y 3 )

Incorrect:
6 x 6 y12

Correct:
6 x5 y 7

Multiply monomial and polynomial
Simplify : −3 x 2 (−2 x 3 + 2 x − 6)

Incorrect:
6 x 2 + 2 x − 6 or 6 x 6 − 6 x 2 + 18 x 2

Correct:
6 x 5 − 6 x 3 + 18 x 2
Polynomials-Multiplying (continued)
How to Multiply
Polynomials


To multiply binomial by a binomial
multiply each term in the first
binomial by each term in the second
binomial. Use “FOIL” method to
assist in remembering which terms
need to be multiplied with which.
Combine like terms.
To multiply a polynomial by another
polynomial multiply each term in
the first polynomial by each term in
the second polynomial. Combine
like terms.
Common Mistakes

Multiply binomials
Simplify : (2 x − 5)(3 x + 2)

Incorrect:
6 x 2 − 10 or 6 x 2 + 4 x − 15 x − 10

Correct:
6 x 2 + 4 x − 15 x − 10 = 6 x 2 − 11x − 10

Multiply polynomials
Simplify : ( x + 2)(3 x 3 + 5 x − 4)

Incorrect:
x(3 x 2 ) + 2(5 x) − 4

Correct:
x(3x 3 ) + x(5 x) − x(4) + 2(3x 3 ) + 2(5 x) − 2(4)
= 3x 4 + 5 x 2 − 4 x + 6 x 3 + 10 x − 8
= 3x 4 + 6 x 3 + 5 x 2 + 6 x − 8
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Polynomials-Special Products
How to Multiply Special
Products
(a + b) 2 = a 2 + 2ab + b 2
Common Mistakes

Multiply (a + b)
2
or (a − b) 2
Simplify : (3 x + 5) 2
(a − b) 2 = a 2 − 2ab + b 2

Incorrect:
9 x 2 + 25 or 3 x 2 + 25

Correct:
(3x) 2 + 2(3x)(5) + (5) 2
(a + b)(a − b) = a 2 − b 2
= 9 x 2 + 30 x + 25
(a + b) = a + 3a b + 3ab + b
3
3
2
2
3

Multiply (a + b)
3
or (a − b) 3
Simplify : (2 x + 3) 3
(a − b) = a − 3a b + 3ab − b
3
3
2
2
3


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Incorrect:
8 x 2 + 27 or 2 x 3 + 27
Correct:
(2 x)3 + 3(2 x) 2 (3) + 3(2 x)(3) 2 + (3) 3
= 8 x 3 + 36 x 2 + 54 x + 27
Polynomials-Dividing
How to Divide Polynomials

To divide a polynomial by a single
term treat the division as a
simplification and reduce each
term to the lowest terms possible.
Common Mistakes

Divide by a monomial
2x2 + 4x
Simplify :
4x
 Incorrect:
2x2 + 4x
≠ 2x2 + 1
4x
2x2 + 4x
or
≠ 2x2
4x

Correct:
2x2 + 4x 2x2 4x x
=
+
= +1
4x
4x 4x 2
2/ x/ ( x + 2) x 2 x
or
= + = +1
2 × 2/ x/
2 2 2
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Polynomials-Long Division
How to Do Long Division of
Polynomials

Long division of polynomials is the
same as regular long division, with
the exception that variables are
included.
Common Mistakes

Long Division
Divide x 2 − 6 x − 12 by x + 2.
 Incorrect:
x−4
2
x + 2 x − 6 x − 16
( x 2 − 6 x − 16) ÷ ( x + 2)
2
x + 2x
≠ ( x − 4) + (−24) /( x + 2)
− 4 x − 16
− 4x − 8
− 24
Correct:
x −8
2
x + 2 x − 6 x − 16

− ( x 2 + 2 x)
− 8 x − 16
− (−8 x − 16)
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0
( x 2 − 6 x − 16) ÷ ( x + 2)
= ( x − 8)