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(male narrator) In this video,
we will begin taking an introductory look
at polynomials.
As we do, we will first lay the vocabulary
needed to know about polynomials.
The first is a term, which is either a number...
such as -3...
a variable...
such as ax...a or x...
or a product...
of numbers and variables...
such as -2x squared, y.
Based on the number of terms in an expression,
we can name the polynomials.
A monomial--the prefix mono, means one--one term.
For example, 3a squared, b cubed, is a monomial.
Binomial--prefix bi-, meaning two--has two terms.
An example of a binomial
would be something like a squared minus b squared.
Notice the two terms are separated by a subtraction.
Addition and subtraction is what separates individual terms.
A trinomial--the prefix tri-, meaning three--has three terms.
An example of a trinomial
would be ax squared, plus bx, plus c.
After trinomial, we group them all together into a polynomial,
which means something with many terms.
In practice, we often call monomials, binomials,
and trinomials, polynomials as well.
An example of a polynomial would be ax squared,
plus by squared, plus cx, plus dy, plus exy, plus f.
Notice the many terms are separated
by addition and subtraction.
Let's take a look at some polynomials
and how we can evaluate them
when we know the value of our variables.
In this expression, we know that x is equal to -2.
This means we will replace each x with a -2:
5 times x, or -2 squared; minus 2; times -2; plus 6.
Now, using order of operations, we can simplify the expression
doing the exponent first:
-2 squared is +4, minus 2, times -2, plus 6.
Next, we multiply,
and as the terms are separate terms,
we can do the multiplication at once:
5 times 4 is 20, -2 times -2 is +4, plus 6.
Finally, we can add the expression from left to right:
20 plus 4, plus 6, to get 30.
This polynomial equals 30 when x is -2.
Let's try one more example where we evaluate a polynomial
when we know the value of the variable.
In this expression, we know the variable x is equal to 4.
This means we will replace each of the x's with a 4,
giving us -4 squared, plus 2, times 4, minus 7.
Notice as I make a substitution,
the number always go in parentheses.
This way, I know when I'm squaring,
does the negative come with--or not--the exponents?
In this case, the exponent is only on the 4,
so that will be always square,
giving us -4 squared, or 16; plus 2; times 4; minus 7.
Multiplying would be our next step,
giving us -16, plus 8, minus 7.
Finally, order of operations
allows us to do the addition and subtraction:
16 plus 8, minus 7, is -15.
By taking the known value
and plugging it in for our variables,
we can evaluate these polynomials at a given value.