Welcome to a lesson
on polynomials of one variable.
Let's begin by looking at examples of algebraic expressions
that are polynomials, given here,
and examples of algebraic expressions
that are not polynomials, given here.
The definition of a polynomial can vary,
based upon the source,
but the terms of a polynomial on one variable
must be a constant
or the product of a constant and a variable
with a non-negative integer exponent.
So looking at our polynomials given here,
notice how we have several constants.
Here, we have the constant 5.
Here, we have the constant 6,
and here we have the constant negative 7.
And now for the remaining variable terms,
let's go ahead and include the coefficients
as well as the exponents on the variable.
So W is equal to 1W to the first.
Negative Z is equal to negative 1 times Z to the first.
And X is equal to 1X to the first.
So notice how the remaining terms
are the product of a constant and a variable
with a non-negative integer exponent.
So again, W is equal to 1 times W to the first.
Here we have negative 7Y to the fourth,
which is equal to negative 7 times Y to the fourth.
Negative Z is equal to negative 1 times Z to the first.
2X to the second is equal to 2 times X to the second.
And X is equal to 1X to the first.
Notice all the terms
fit into these two categories.
Now let's look at the examples of the expressions
that are not polynomials.
First, we have 5 divided by X.
Notice this is not a product of a constant and a variable.
Here we have 5 divided by X, so we have a quotient.
We can also write this in the form of
5 times X to the power of negative 1.
Notice in this form the exponent
is not a non-negative integer.
Next we have negative 7Y to the two-thirds.
Because the exponent is two-thirds here,
and two-thirds is not a non-negative integer,
this is not a polynomial.
And finally, in our last expression,
the first and the third terms do satisfy the conditions
for a polynomial, but the second term doesn't
and therefore this is not a polynomial.
The square root of X
is equal to X raised to the power of one-half,
and once again this exponent is not a non-negative integer.
Now let's look at some additional definitions.
A monomial is a polynomial consisting of one term.
So for example here we have a polynomial
consisting of four terms.
But if the polynomial had one term,
for example if the polynomial was just 12X to the fifth,
this would be a monomial.
A binomial is a polynomial consisting of two terms.
So for example if we had a polynomial
that was 12X to the fifth,
minus 2X to the second,
this would be a binomial.
A trinomial is a polynomial consisting of three terms.
So for example, if we had a polynomial
like 12X to the fifth,
minus 2X to the second plus X,
this would be an example of a trinomial.
Next, a leading term is a term that contains
the highest power of the variable in the polynomial.
So using this as an example,
notice the highest power on X is 5,
and therefore the leading term would be 12X to the fifth.
The leading coefficient
is the coefficient of the leading term.
Since 12X to the fifth is equal to 12 times X to the fifth,
the leading coefficient would be 12.
The constant term is the term with no variable factor.
So again looking at this polynomial here,
notice how the term negative 7 does not have a factor of X,
and therefore negative 7 would be the constant term.
And then finally the degree of a polynomial
is the highest exponent on the variable in the polynomial.
Again, the highest exponent on X is 5,
therefore this is a degree 5 polynomial.
Now let's look at example 1.
Here we're given different polynomials.
We're asked to name them,
find the leading coefficient,
find the constant term
and then finally give the degree.
So again the name is based upon the number of terms,
so this polynomial has three terms,
so we call it a trinomial.
This polynomial has four terms.
Once it has more than three terms,
we simply call it a polynomial.
This polynomial has three terms.
We have another trinomial.
This polynomial has two terms, which is a binomial.
And here we have one term, so we have a monomial.
Next, we're asked to find the leading coefficient,
which is the coefficient of the leading term.
Remember, the leading term
is the term that has the highest exponent on the variable.
Notice here the leading term is 24A to the sixth,
because the highest exponent on A is 6
and therefore the leading coefficient would be 24.
In this polynomial, the highest exponent on M is 3,
so this would be the leading term,
and therefore the leading coefficient is 2.
Need to be careful on this example.
The terms are not in descending order.
The highest exponent on X
is 3, so this would be the leading term.
Notice how the coefficient would be 1.
So the leading coefficient is 1.
In our binomial,
since this is X to the first,
the leading term is negative 2X,
and therefore the leading coefficient is negative 2.
Here we only have one term,
so this is the leading term,
and therefore the leading coefficient is 4.
Next we're asked to give the constant term,
which is the term that does not contain a variable factor.
So the constant term in this polynomial would be 5.
The constant term of this polynomial would be negative 8
because of the subtraction.
The constant term here is negative 7.
The constant term here is positive 4.
Now for our monomial, it looks like
there is no constant term, but of course we could
write plus 0, where 0 is a constant,
so the constant term would be 0.
Finally, we're asked to find the degree.
The degree of a polynomial in one variable
is the highest exponent on a variable in the polynomial.
So notice here the highest exponent on A is 6,
so this is a degree 6 polynomial.
The highest exponent on M here is 3,
so we have a degree 3 polynomial.
The highest exponent on X here is 3,
so we have another degree 3 polynomial.
The highest exponent on X here is 1,
so we have a degree 1 polynomial.
And here the highest exponent on X is 3,
so we have a degree 3 polynomial.
Okay, I hope you found this helpful.