11.2.1: Introduction to Single Variable Polynomials
- Page ID
- 67619
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- Evaluate a polynomial for given values of the variable.
- Simplify polynomials by collecting like terms.
Introduction
Algebraic expressions are created by combining numbers and variables using arithmetic operations: addition, subtraction, multiplication, division, and exponentiation. Using all but division, you can create an expression called a polynomial by adding or subtracting terms. Polynomials are very useful in applications from science and engineering to business. Monomials (and polynomials in general) may have more than one variable, but in this unit, you will only work with single variable polynomials.
Monomials
The basic building block of a polynomial is a monomial. A monomial is one term and can be a number, a variable, or the product of a number and variables with an exponent. The number part of the term is called the coefficient.
The coefficient can be any real number, including 0. The exponent of the variable must be a whole number: 0, 1, 2, 3, and so on. A monomial cannot have a variable in the denominator or a negative exponent.
The value of the exponent is the degree of the monomial. Remember that a variable that appears to have no exponent really has an exponent of 1. And a monomial with no variable has a degree of 0. (\(\ x^{0} \) has the value of 1 if \(\ x \neq 0\). So a number such as 3 could be written as \(\ 3 x^{0}\), if \(\ x \neq 0\). When \(\ x \neq 0\) it can be written as \(\ 3 x^{0}=3 \cdot 1=3\).)
Identify the coefficient, variable, and exponent of the monomial \(\ \frac{3}{5} k^{8}\).
Solution
The variable is \(\ k\).
The exponent of \(\ k\) is 8.
The coefficient of \(\ k^{8}\) is \(\ \frac{3}{5}\).
Identify the coefficient, variable, and exponent of \(\ x\).
Solution
| The exponent of \(\ x\) is 1. | \(\ x=x^{1}\), so the exponent is 1. |
| The coefficient of \(\ x\) is 1. | \(\ x=1 x^{1}\) so the coefficient is also 1. |
The variable is \(\ x\).
Identify the coefficient, variable, and exponent of \(\ 3y\).
- The variable is \(\ y\), the exponent is 3, and the coefficient is 1.
- The variable is \(\ y\), the exponent is 0, and the coefficient is 3.
- The variable is \(\ y\), the exponent is 1, and the coefficient is 3.
- The variable is \(\ y\), the exponent is 3, and the monomial has no coefficient.
- Answer
-
- The variable is \(\ y\), the exponent is 3, and the coefficient is 1.
Incorrect. The exponent is the power of the variable and the coefficient is the number before the variable. The coefficient in this case is 3, and the exponent is 1 because \(\ 3 y=3 y^{1}\).
- The variable is \(\ y\), the exponent is 0, and the coefficient is 3.
Incorrect. The coefficient is 3, but the exponent is 1 because \(\ 3 y=3 y^{1}\). An exponent of 0 would be \(\ 3 y^{0}\), and since \(\ y^{0}=1\) when \(\ y \neq 0\), this would be equal to 3 (when \(\ y \neq 0\)).
- The variable is \(\ y\), the exponent is 1, and the coefficient is 3.
Correct. The coefficient is 3, the variable is \(\ y\), and the exponent is 1 because \(\ 3 y=3 y^{1}\).
- The variable is \(\ y\), the exponent is 3, and the monomial has no coefficient.
Incorrect. The exponent is the power of the variable and the coefficient is the number before the variable. The coefficient in this case is 3, and the exponent is 1 because \(\ 3 y=3 y^{1}\).
- The variable is \(\ y\), the exponent is 3, and the coefficient is 1.
Naming and Writing Polynomials
A polynomial is a monomial or the sum or difference of two or more polynomials. Each monomial is called a term of the polynomial.
Some polynomials have specific names indicated by their prefix.
monomial - is a polynomial with exactly one term (“mono” - means one)
binomial - is a polynomial with exactly two terms (“bi” - means two)
trinomial - is a polynomial with exactly three terms (“tri” - means three)
The word “polynomial” has the prefix, “poly,” which means many. However, the word polynomial can be used for all numbers of terms, including only one term.
Because the exponent of the variable must be a whole number, monomials and polynomials cannot have a variable in the denominator.
Polynomials can be classified by the degree of the polynomial. The degree of a polynomial is the degree of its highest-degree term. So the degree of \(\ 2 x^{3}+3 x^{2}+8 x+5\) is 3.
A polynomial is said to be written in standard form when the terms are arranged from the highest degree to the lowest degree. When it is written in standard form, it is easy to determine the degree of the polynomial.
The tables below illustrate some examples of monomials, binomials, trinomials, and other polynomials. They are all written in standard form.
| 15 |
| \(\ \frac{1}{2} x\) |
| \(\ -4 y^{3}\) |
| \(\ 16 n^{4}\) |
| \(\ 3 y+13\) |
| \(\ 4 p-7\) |
| \(\ 3 x^{2}+\frac{5}{8} x\) |
| \(\ 14 y^{3}+3 y\) |
| \(\ x^{3}-x^{2}+1\) |
| \(\ 3 x^{2}+2 x-9\) |
| \(\ 3 y^{3}+y^{2}-2\) |
| \(\ a^{7}+2 a^{5}-3 a^{3}\) |
| \(\ 5 x^{4}+3 x^{3}-6 x^{2}+2 x\) |
| \(\ \frac{1}{3} x^{5}-2 x^{4}+\frac{2}{9} x^{3}-x^{2}+4 x-\frac{5}{6}\) |
| \(\ 3 t^{3}-3 t^{2}-3 t-3\) |
| \(\ q^{7}+2 q^{5}-3 q^{3}+q\) |
When the coefficient of a polynomial term is 0, you usually do not write the term at all (because 0 times anything is 0, and adding 0 doesn’t change the value). The last binomial in the binomial table, \(\ 14 y^{3}+3 y\), could be written as a trinomial, \(\ 14 y^{3}+0 y^{2}+3 y\).
A term without a variable is called a constant term, and the degree of that term is 0. For example 13 is the constant term in \(\ 3 y+13\). You would usually say that \(\ 14 y^{3}+3 y\) has no constant term or that the constant term is 0.
Which of the following expressions are polynomials?
\(\ 2 x^{4}-3 x^{3}\)
\(\ 14\)
\(\ \frac{x+1}{x}\)
- Only \(\ 2 x^{4}-3 x^{3}\) and \(\ \frac{x+1}{x}\) are polynomials.
- Only \(\ 2 x^{4}-3 x^{3}\) and 14 are polynomials.
- Only \(\ 2 x^{4}-3 x^{3}\) is a polynomial.
- None of the expressions is a polynomial.
- Answer
-
- Only \(\ 2 x^{4}-3 x^{3}\) and \(\ \frac{x+1}{x}\) are polynomials.
Incorrect. \(\ \frac{x+1}{x}\) is not a polynomial because it has a variable in the denominator. The correct answer is \(\ 2 x^{4}-3 x^{3}\) and 14.
- Only \(\ 2 x^{4}-3 x^{3}\) and 14 are polynomials.
Correct. \(\ \frac{x+1}{x}\) is not a polynomial because it has a variable in the denominator.
- Only \(\ 2 x^{4}-3 x^{3}\) is a polynomial.
Incorrect. 14 is also a monomial, a type of polynomial. The correct answer is \(\ 2 x^{4}-3 x^{3}\) and 14.
- None of the expressions is a polynomial.
Incorrect. \(\ x^{4}-3 y^{3}\) and 14 are polynomials. \(\ \frac{x+1}{x}\) is not a polynomial because it has a variable in the denominator.
- Only \(\ 2 x^{4}-3 x^{3}\) and \(\ \frac{x+1}{x}\) are polynomials.
Evaluating Polynomials
You can evaluate polynomials just as you have been evaluating expressions all along. To evaluate an expression for a value of the variable, you substitute the value for the variable every time it appears. Then use the order of operations to find the resulting value for the expression.
Evaluate \(\ 3 x^{2}-2 x+1\) and \(\ x=-1\).
Solution
| \(\ 3(-1)^{2}-2(-1)+1\) | Substitute -1 for each \(\ x\) in the polynomial. |
| \(\ 3(1)-2(-1)+1\) | Following the order of operations, evaluate exponents first. |
| \(\ 3+(-2)(-1)+1\) | Multiply 3 times 1, and then multiply -2 times -1. |
| \(\ 3+2+1\) | Change the subtraction to addition of the opposite. |
| \(\ 3 x^{2}-2 x+1=6, \text { for } x=-1\) | Find the sum. |
Evaluate \(\ -\frac{2}{3} p^{4}+2 p^{3}-p\) for \(\ p=3\)
Solution
| \(\ -\frac{2}{3}(3)^{4}+2(3)^{3}-3\) | Substitute 3 for each \(\ p\) in the polynomial. |
| \(\ -\frac{2}{3}(81)+2(27)-3\) | Following the order of operations, evaluate exponents first and then multiply. |
| \(\ -54+54-3\) | Add and then subtract to get -3. |
\(\ -\frac{2}{3} p^{4}+2 p^{3}-p=-3, \text { for } p=3\)
Evaluate \(\ 3 x^{3}-2 x^{2} \text { for } x=-2\).
- \(\ -24-2 x^{2}\)
- \(\ -32\)
- \(\ -16\)
- \(\ 16\)
- Answer
-
- \(\ -24-2 x^{2}\)
Incorrect. \(\ x=-2\) for all appearances of \(\ x\) in the expression. You must substitute -2 into both instances: \(\ 3(-2)^{3}-2(-2)^{2}\). The correct answer is -32.
- \(\ -32\)
Correct. Since \(\ x=-2\): \(\ 3 x^{3}-2 x^{2}=3(-2)^{3}-2(-2)^{2}=3(-8)-2(4)=-24-8=-32\).
- \(\ -16\)
Incorrect. You may have forgotten that \(\ (-2)^{2}\) is positive, or made another mistake in your calculations. Since \(\ x=-2\): \(\ 3 x^{3}-2 x^{2}=3(-2)^{3}-2(-2)^{2}=3(-8)-2(4)=-24-8=-32\).
- \(\ 16\)
Incorrect. You may have forgotten that \(\ (-2)^{3}\) is negative (the product of two negative values is positive, so multiplying by a third negative gives a negative product). Since \(\ x=-2\): \(\ 3 x^{3}-2 x^{2}=3(-2)^{3}-2(-2)^{2}=3(-8)-2(4)=-24-8=-32\).
- \(\ -24-2 x^{2}\)
Simplifying Polynomials
A polynomial may need to be simplified. One way to simplify a polynomial is to combine the like terms if there are any. Two or more terms in a polynomial are like terms if they have the same variable (or variables) with the same exponent. For example, \(\ 3 x^{2}\) and \(\ -5 x^{2}\) are like terms: They both have \(\ x\) as the variable, and the exponent is 2 for each. However, \(\ 3 x^{2}\) and \(\ 3x\) are not like terms, because their exponents are different.
Here are some examples of terms that are like and some that are unlike.
| Monomials | Terms | Explanation |
| \(\ \begin{array}{l} 3 x \\ 14 x \end{array}\) |
like | same variables with same exponents |
| \(\ \begin{array}{l} 16 z^{2} \\ -5 z^{2} \end{array}\) |
like | same variables with same exponents |
| \(\ \begin{array}{l} 3x \\ 5y \end{array}\) |
unlike | different variables (although the same exponents) |
| \(\ \begin{array}{l} -3z\\ -3z^2 \end{array}\) |
unlike | same variables but with different exponents |
Which of these terms are like terms?
\(\ 7 x^{3}, 7 x, 7 y,-8 x^{3}, 9 y,-3 x^{2}, 8 y^{2}\)
Solution
| \(\ \begin{array}{l} x: 7 x^{3}, 7 x,-8 x^{3},-3 x^{2} \\ y: 7 y, 9 y, 8 y^{2} \end{array}\) |
Like terms must have the same variables, so first identify which terms use the same variables. |
| The \(\ x\) terms \(\ 7 x^{3}\) and \(\ -8 x^{3}\) have the same exponent.he \(\ y\) terms \(\ 7 y\) and \(\ 9 y\) have the same exponent. | Like terms must also have the same exponents. Identify which terms with the same variables also use the same exponents. |
\(\ 7 x^{3}\) and \(\ -8 x^{3}\) are like terms.
\(\ 7 y\) and \(\ 9 y\) are like terms.
Which of these are like terms?
\(\ -3 a, 3 a^{2}, 8 b,-3 b^{3}, 8 a, 14 b^{2}, 9 a\)
- Only \(\ -3 a\), \(\ 8 a\), and \(\ 9a\) are like terms.
- Only \(\ 8a\) and \(\ 9a\) are like terms.
- \(\ 8b\), \(\ -3 b^{3}\), and \(\ 14 b^{2}\) are like terms, and \(\ -3 a\), \(\ 3 a^{2}\), \(\ 8 a\), and \(\ 9a\) are like terms.
- \(\ 3 a^{2}\) and \(\ 14 b^{2}\) are like terms, and \(\ -3 a\), \(\ 8 b\), \(\ 8a\), and \(\ 9a\) are like terms.
- Answer
-
- Only \(\ -3 a\), \(\ 8 a\), and \(\ 9a\) are like terms.
Correct. These terms have the same variable, \(\ a\), with an exponent of 1.
- Only \(\ 8a\) and \(\ 9a\) are like terms.
Incorrect. \(\ -3 a\) also is a like term with \(\ 8a\) and \(\ 9 a\), because it has the same variable with the same exponent.
- \(\ 8b\), \(\ -3 b^{3}\), and \(\ 14 b^{2}\) are like terms, and \(\ -3 a\), \(\ 3 a^{2}\), \(\ 8 a\), and \(\ 9a\) are like terms.
Incorrect. Each group has the same variable, but their exponents are different. The only terms with the same variable and exponent are \(\ -3 a\), \(\ 8 a\), and \(\ 9 a\).
- \(\ 3 a^{2}\) and \(\ 14 b^{2}\) are like terms, and \(\ -3 a\), \(\ 8 b\), \(\ 8a\), and \(\ 9a\) are like terms.
Incorrect. Each group has the same exponent, but their variables are different. The only terms with the same variable and exponent are \(\ -3 a\), \(\ 8 a\), and \(\ 9a\).
- Only \(\ -3 a\), \(\ 8 a\), and \(\ 9a\) are like terms.
You can use the distributive property to simplify the sum of like terms. Recall that the distributive property of addition states that the product of a number and a sum (or difference) is equal to the sum (or difference) of the products.
\(\ 2(3+6)=2(3)+2(6)\)
Both expressions equal 18. So you can write the expression in whichever form is the most useful.
Let’s see how we can use this property to combine like terms.
Simplify \(\ 3 x^{2}-5 x^{2}\).
Solution
| \(\ 3\left(x^{2}\right)-5\left(x^{2}\right)\) | \(\ 3 x^{2}\) and \(\ 5 x^{2}\) are like terms. |
| \(\ (3-5)\left(x^{2}\right)\) | We can rewrite the expression as the product of the difference. |
| \(\ (-2)\left(x^{2}\right)\) | Calculate \(\ 3-5\). |
| \(\ 3 x^{2}-5 x^{2}=-2 x^{2}\) | Write the difference of \(\ 3-5\) as the new coefficient. |
You may have noticed that combining like terms involves combining the coefficients to find the new coefficient of the like term. You can use this as a shortcut.
Simplify \(\ 6 a^{4}+4 a^{4}\).
Solution
| \(\ 6 a^{4}+4 a^{4}\) | Notice that both terms have a number multiplied by \(\ a^{4}\). This makes them like terms. |
| \(\ (6+4)\left(a^{4}\right)\) | Combine the coefficients, 6 and 4. |
| \(\ (10)\left(a^{4}\right)\) | Calculate the sum. |
| \(\ 6 a^{4}+4 a^{4}=10 a^{4}\) | Write the sum as the new coefficient. |
When you have a polynomial with more terms, you have to be careful that you combine only like terms. If two terms are not like terms, you can’t combine them.
Simplify \(\ 3 x^{2}+3 x+x+1+5 x\).
Solution
| \(\ 3x\), \(\ x\), and \(\ 5x\) are like terms | First identify which terms are like terms: only \(\ 3x\), \(\ x\), and \(\ 5x\) are like terms. |
| \(\ 3 x^{2}+3 x+x+1+5 x=3 x^{2}+(3 x+x+5 x)+1\) | Use the commutative and associative properties to group the like terms together. |
| \(\ 3 x^{2}+(3+1+5) x+1\) | Add the coefficients of the like terms. Remember that the coefficient of \(\ x\) is 1 because \(\ x=1 x\). |
| \(\ 3 x^{2}+(9) x+1\) | |
| \(\ 3 x^{2}+3 x+x+1+5 x=3 x^{2}+9 x+1\) | Write the sum as the new coefficient. |
Simplify by combining like terms.
\(\ -3 a+3 a^{2}+8 a+9 a-3\)
- \(\ 3 a^{2}+17 a-3\)
- \(\ 3 a^{2}+14 a-3\)
- \(\ 17 a-3\)
- \(\ 17 a-3 a-3\)
- Answer
-
- Incorrect. There are three terms with \(\ a\) (remember that \(\ a^{1}\)) is written as \(\ a\)): \(\ -3 a\), \(\ 8a\), and \(\ 9a\). Combining them gives \(\ -3 a+8 a+9 a=(-3+8+9) a=14 a\), so the simplified polynomial is \(\ 3 a^{2}+14 a-3\).
- Correct. There are three terms with \(\ a\) (remember that \(\ a^{1}\) is written as \(\ a\)): \(\ -3a\), \(\ 8a\), and \(\ 9a\). Combining them gives \(\ -3 a+8 a+9 a=(-3+8+9) a=14 a\), so the simplified polynomial is \(\ 3 a^{2}+14 a-3\).
- Incorrect. You may have incorrectly combined the first two terms, \(\ -3 a+3 a^{2}\). These are not like terms, so you can’t combine them. There are three terms with \(\ a\) (remember that \(\ a^{1}\) is written as \(\ a\)): \(\ -3a\), \(\ 8a\), and \(\ 9a\). Combining them gives \(\ -3 a+8 a+9 a=(-3+8+9) a=14 a\), so the simplified polynomial is \(\ 3 a^{2}+14 a-3\).
- Incorrect. There are three terms with \(\ a\) (remember that \(\ a^{1}\) is written as \(\ a\)): \(\ -3a\), \(\ 8a\), and \(\ 9a\). Combining them gives \(\ -3 a+8 a+9 a=(-3+8+9) a=14 a\), so the simplified polynomial is \(\ 3 a^{2}+14 a-3\).
Summary
Polynomials are algebraic expressions that contain any number of terms combined by using addition or subtraction. A term is a number, a variable, or a product of a number and one or more variables with exponents. Like terms (same variable or variables raised to the same power) can be combined to simplify a polynomial. The polynomials can be evaluated by substituting a given value of the variable into each instance of the variable, then using order of operations to complete the calculations.