A polynomial term is a term where the variable contains a non-negative integer power. (The non-negative integers consist of the numbers and so on.)
Some examples of polynomial terms include
Let's go through each one:
is a polynomial term because the power of the variable is , a non-negative integer.
is a polynomial term because the power of the variable is , which is also a non-negative integer. The coefficient of isn't important.
is a polynomial term because it can be thought of as So the power of the variable is , which is a non-negative integer.
Finally, is a polynomial term because it can be expressed as So the power of the variable is , which is a non-negative integer.
On the other hand, the following expressions are not polynomial terms because their powers are not non-negative integers:
To see why, let's go through each one:
is not a polynomial term because the power of the variable is , which is negative.
is not a polynomial term because the power of the variable is , which is not an integer.
is not a polynomial term because it can be written as So the power of the variable is , which is not an integer.
Finally, is not a polynomial term because it can be written as So the power of the variable is , which is negative.
Which of the following is a polynomial term?
Remember that a polynomial term is a term where the variable contains a non-negative integer power. (The non-negative integers consist of and so on.)
With that in mind, let's take a look at each term.
Expression I is not a polynomial term since Here, is raised the power of which is not an integer.
Expression II is a polynomial term since the variable is raised the power of , which is indeed a non-negative integer number.
Expression III is not a polynomial because Here, is raised the power of which is negative.
In conclusion, only expression II is a polynomial term.
Which of the following is a polynomial term?
- $-x$
- $3\sqrt x$
- $\dfrac 1 {x^2}$
|
a
|
I and III only |
|
b
|
I and II only |
|
c
|
I only |
|
d
|
III only |
|
e
|
II and III only |
Which of the following is a polynomial term?
- $2x$
- $2 \sqrt{x}$
- $\dfrac{1}{2}x^2$
|
a
|
I, II and III |
|
b
|
I and III only |
|
c
|
III only |
|
d
|
I only |
|
e
|
I and II only |
Which of the following expressions is a polynomial?
Let's look at each statement in turn.
Expression I is a polynomial because all terms are polynomial terms.
Expression II is a polynomial because both terms are polynomial terms. Even though has a square root, the square root is not on the variable. It can be written as in which case the variable is raised to a non-negative integer power.
Expression III is not a polynomial because is not a polynomial term. It is equivalent to in which case the exponent on the variable is not an integer.
In conclusion, only expressions I and II are polynomials.
Which of the following is NOT a polynomial?
|
a
|
$t+\sqrt{2}$ |
|
b
|
$\sqrt{t}+1$ |
|
c
|
$t^2-1$ |
|
d
|
$\dfrac{t}{2}-1$ |
|
e
|
$\dfrac{t}{\sqrt{2}}+1$ |
Which of the following is a polynomial?
- $-\dfrac x 3 + \sqrt{2} x^2$
- $2x^4 - 3$
- $3x^{2/3} - \sqrt{3} x$
|
a
|
I only |
|
b
|
II and III only |
|
c
|
II only |
|
d
|
I and III only |
|
e
|
I and II only |
To evaluate a polynomial at a specific value of we substitute the value of into the polynomial and simplify.
For example, to evaluate the polynomial at we substitute into the expression and simplify:
Evaluate the polynomial at
We substitute into the polynomial and simplify:
Evaluate the polynomial $-2x^3-3$ at $x=2.$
|
a
|
$-13$ |
|
b
|
$3$ |
|
c
|
$1$ |
|
d
|
$-19$ |
|
e
|
$-16$ |
Evaluate the polynomial $x^2-2x+1$ at $x=5.$
|
a
|
$21$ |
|
b
|
$16$ |
|
c
|
$5$ |
|
d
|
$11$ |
|
e
|
$1$ |
Given the polynomial what is
We substitute into the expression for and simplify:
Therefore,
Given the polynomial $g(x)=\dfrac{1}{2}x^2+2x-5,$ $g(-4)=$
|
a
|
$-9$ |
|
b
|
$11$ |
|
c
|
$-11$ |
|
d
|
$-5$ |
|
e
|
$-21$ |
Given the polynomial $f(x)=3x^2-2x+1,$ $f(2)=$
|
a
|
$17$ |
|
b
|
$16$ |
|
c
|
$2$ |
|
d
|
$1$ |
|
e
|
$9$ |