An Introduction to Polynomials
Copyright Scott Storla 2015
Some Vocabulary for Polynomials
Copyright Scott Storla 2015
Coefficient
3x  5
Variable term
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Constant term
Definition – Polynomial
A polynomial in x is a single term, or a sum of terms,
where each term is a variable term or a constant.
Every variable term has a coefficient, the variable x,
and an exponent of x that is a natural number.
Example: 2x 3  3 x  5
Notice a polynomial is a sum.
You should think of
3x 2  x  4
as
3x 2  1x1  4
Copyright Scott Storla 2015
Special names for the number of terms.
One term
3k
Two terms
3k  7
monomial
binomial
Three terms 3k  n  7 trinomial
Copyright Scott Storla 2015
1. Write the polynomial as a sum with all
coefficients and exponents explicit.
2. Discuss the polynomial in both general and
specific terms.
n  14
1n1  14
Copyright Scott Storla 2015
1. Write the polynomial as a sum with all
coefficients explicit.
2. Discuss the polynomial in both general and
specific terms.
y  5
1y 1  5
Copyright Scott Storla 2015
The degree of a term
For each variable term use the exponent to
decide on the degree of the term.
8a 4  7a3  5a2  1
The degree of a polynomial
The degree of the entire polynomial is the same as
the degree of the term with the largest exponent.
8a 4  7a3  5a2  1
Copyright Scott Storla 2015
Standard Form
The terms of the polynomial are written in
decreasing order of degree from left to right.
b  5  7b3  2b 2
Not in standard form
7b3  2b 2  b  5
Standard form
To write a polynomial in standard form we imagine
all operations are addition and all coefficients are
explicit, then we use the commutative property to
rearrange the terms, last we rewrite all explicit
coefficients implicitly.
b  5  7b3  2b 2
1b  5  7b3  2b 2
7b3  2b 2  1b  5
7b3  2b 2  b  5
Copyright Scott Storla 2015
Standard Form
In practice people rearrange the terms of a
polynomial “in their head”.
Write each polynomial in standard form.
2 x  15  5 x 4  x 2  x 3
5 x 4  x 3  x 2  2 x  15
9 y 7  12y  y 3  y 9  15 y 2
 y 9  9 y 7  y 3  15 y 2  12y
7k 5  5k 7  8k 3  3k 8  6  4k 2  2k 4
3k 8  5k 7  7k 5  2k 4  8k 3  4k 2  6
Copyright Scott Storla 2015
1. Write the polynomial in standard form
2. Discuss the polynomial in general terms.
3. Discuss the polynomial term by term.
3  2x 2  x
2x 2  x  3
15  15y 3
15 y 3  15
6n 5  5n 6  4n  5  6n 2
5n 6  6n 5  6n 2  4n  5
Copyright Scott Storla 2015
1. Write the polynomial in standard form
2. Discuss the polynomial in general terms.
3. Discuss the polynomial term by term.
7k 5  5k 7  8k 3  3k 8  6  4k 2  2k 4
3k 8  5k 7  7k 5  2k 4  8k 3  4k 2  6
Copyright Scott Storla 2015
Multivariable or “mixed” terms
With multivariable terms the degree of the term
is the sum of the individual exponents. We
don’t actually add the exponents.
ab
A second degree term
2 xy 4
3 x 2 y 3 z2
A fifth degree term
A seventh degree term
Copyright Scott Storla 2015
Multivariable or “mixed” terms
For standard form, terms of equal degree can
be written in any order but often decisions are
made using alphabetical order.
b 2a is often rewritten ab 2 but a 2 b is left alone.
2y 2  2 x 2 is usually rewritten 2 x 2  2 y 2
Even though 7 xy and 5 x 2 are both second
degree terms, they are usually written in
the order 5 x 2  7 xy .
Copyright Scott Storla 2015
1. Write the polynomial in standard form
2. Discuss the polynomial in general terms.
3. Discuss the polynomial term by term.
3 xy 2  2 x 2 y  x 2 y 2
 x 2 y 2  2 x 2 y  3 xy 2
7ab 2  2a3  5a 2 b  b3
2a3  5a 2 b  7ab 2  b3
12 j 2 k  2  2 jk  15k 4
15k 4  12 j 2 k  2 jk  2
Copyright Scott Storla 2015
Some Vocabulary for Polynomials
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Adding and Subtracting Polynomials
Copyright Scott Storla 2015
Only like terms can be added or subtracted.
Terms are like if, in general, they’re
counting the same sized unit.
Copyright Scott Storla 2015
Like Polynomial Terms
Polynomial terms in one variable are like if the
variable has the same exponent. Constant
terms are also considered like.
6 y ,  11y
6c 5 ,  11c 5
6k 2 ,  11k
Are like terms
Are like terms
Are not like terms
Copyright Scott Storla 2015
Decide on the like terms
t  5  t3  7  t
4 x 2  2 x 2  7 x  x 2
1
12  y  8 y 3  7 y 2  y 3
2
5  k 3  4k  1  k 2
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Simplify
4 x 2  3 x 3  4  x 3  2x 2
2x 3  2x 2  4
Copyright Scott Storla 2015
Simplify
4x  3  4  x  2
5x 1
6 y  y  9  4  2  5 y
0y 3
3
3 x  3  4 x  x  11  7 x  8
x 0
x
Copyright Scott Storla 2015
Simplify
4x 2  3  4  x 2  2
5x 2  1
6 y 3  y 2  9  4 y 2  2  5 y 3
y 3  3y 2  7
5 p7  7 p 4  4 p 4  p7  4 p7  3 p 4
6p 4
2y 2  5 y 3  4 y 2  y  9 y 3
4 y 3  6 y 2  y
3 x 3  3 x 2  4 x 2  x  11x 3  7 x 2  8 x 3
x
Copyright Scott Storla 2015
Simplify
3 x 2y  4 yx  yx 2  8 xy
2 x 2 y  4 xy
2 x  15  4 xy  2  xy  11x
3 xy  9 x  13
x 2y  6 y 2 x 2  xy 2  2 x 2y 2
4x 2 y 2  x 2 y  xy 2
3a2b  5ba2  6a2b2  3  9b2  14b2a2
20a 2 b 2  2a 2 b  9b 2  3
4i 2 j  4i 2 j  j 2 i  7i 2 j 2  7 ji 2  15i 2 j  5 j 2 i 2
2i 2 j 2  ij 2
Copyright Scott Storla 2015
Adding and Subtracting Polynomials
Copyright Scott Storla 2015