Intro to Polynomials
Hey, guys! Welcome to this Mometrix introduction video to polynomials.
So, polynomial comes from poly, which means “many,” and nomial, meaning “term.” So, really it just says “many terms.” Polynomials can have exponents like the 4 in \(x^{4}\), they can have variables like x and y, and they can have constants like 7,000, -3 million, \(\frac{1}{4}\), 1, etc.. You can even combine all of these using addition, subtraction, multiplication and division. However, a polynomial can never be divided by a variable, only by a constant.
For example:
\(\frac{-28}{7x+1}\)
This is not a polynomial, because there is a variable in the denominator.
Now, we have to be careful and remember all of our algebra tricks. Take a look at this.
\(\frac{-28}{7x^{-2}+1}\)
Because the exponent in the bottom has a negative exponent, we know that, when simplified, the exponent is actually moved to the top, which means that this is a polynomial.
Let’s look at another example:
\(290yx^{-5}\)
This is not a polynomial, because when simplified the variable y is moved from the numerator to the denominator, because of the negative exponent; and a polynomial cannot have a variable in the denominator.
Now, try this one:
\(7xy^{24}+28x-7y^{23}-3\)
Polynomial or not?
This one is a polynomial! It does not have a variable in the denominator.
Another thing that is important to know, is that a polynomial can also just have one term. For instance, 5 alone is a polynomial. This may seem confusing, given that polynomial actually says “many terms,” so I want to make that clear.
Polynomials with 1, 2, or 3 terms all have special names. The name for a polynomial with 1 term is a monomial, a polynomial with 2 terms is a binomial, and a polynomial with 3 terms, as you might guess, is a trinomial. There is also the quadrinomial with 4 terms, and the quintinomial with 5 terms; but those are not often used.
A polynomial can have as many terms as you need, but it cannot have an infinite number of terms.
Polynomials are really great in math because they are easy to work with. If you add polynomials together, you are still left with a polynomial. If you multiply polynomials together, you still have a polynomial.
You can divide polynomials; however, the result may not be a polynomial.
Also, polynomials with one variable are great for graphing because their lines are always smooth and continuous.
The last thing you need to know about a polynomial is the standard way to write a polynomial. The standard way to write a polynomial is in descending order, with the highest degree first (farthest to the left), and the lowest degree last. Degrees, in this sense, refer to the number in the exponent.
So, to put the polynomial, \(7x-7+24x^{3}+1x^{7}\), you would move the term with the highest number in the exponent to the front. In this case, you would move \(1x^{7}\) to the front, and so on. Once you have completed, the standard form of this polynomial is \(1x^{7}+24x^{3}+7x-7\).
I hope this video helped you to understand polynomials a little better. See you next time!
Polynomial Practice Questions
Which expression is not a polynomial?
A polynomial is an expression that can have many terms with variables, coefficients, and positive exponents. These terms can be added, subtracted, multiplied, and divided. When simplified, a polynomial can’t have a variable in its denominator, and it can’t contain a negative exponent. The expression \(5x^2-8x\) has the variable x in its denominator, so answer C is not a polynomial. Answers A, B, and D are all polynomials. Although answer A has a variable in its denominator, it can be simplified to \(3x^2-5x\), which is a polynomial.
Which statement is true?
A polynomial can have an infinite number of terms.
A polynomial can contain a negative exponent.
An expression with one term can be a polynomial.
A polynomial with two terms is called a trinomial.
Although the word polynomial means many terms, a polynomial can also have exactly one term. A polynomial with one term is called a monomial. Answers A, B, and D are false. Polynomials can have as many terms as you need, but not an infinite number of terms. When simplified, polynomials cannot contain negative exponents, or variables in the denominator. A polynomial with two terms is called a binomial.
Which statement is false?
If you add two polynomial expressions, the result is a polynomial.
If you divide two polynomial expressions, the result is a polynomial.
If you multiply two polynomial expressions, the result is a polynomial.
Polynomials with one variable graph as a smooth and continuous line.
When dividing polynomials, the result may not be a polynomial.
The length of a rectangle is written as \(8x+5+x^2\). Its width is written as \(4+6x\). Which answer option shows both polynomials written in standard form?
Length: \(x^2+8x+5\)
Width: \(6x+4\)
Length: \(5+8x+x^2\)
Width: \(4+6x\)
Length: \(8x+x^2+5\)
Width: \(4x+6\)
Length: \(x^2-8x-5\)
Width: \(6+4x\)
Polynomials are written in descending order, with the highest degree first and the lowest degree last. Since the highest degree in the length is 2, the term \(x^2\) must come first in the expression. This is followed by \(8x\), and 5 is written last. Since the highest degree in the width is 1, the term \(6x\) must come first in the expression, followed by 4.
Allison and Christina are racing to see who can run faster around a track. Since Christina had more time to practice, she agreed to give Allison a head start of 50 meters. After x seconds, Allison is \(8x+50\) meters from the starting line. What type of polynomial is Allison’s distance from the starting line, \(8x+50\)?
Quadrinomial
Trinomial
Binomial
Monomial
A binomial is a polynomial that contains two terms. In the expression \(8x+50\), the two terms are \(8x\) and 50.