Introduction to Single Variable Polynomials
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 polynomialA monomial or the sum or difference of two or more monomials. 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.
The basic building block of a polynomial is a monomial A polynomial with exactly one term. `4x`, `-5y^2`, and `6` are all examples of monomials. . 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 coefficientA number that multiplies a variable..
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 degreeThe value of an exponent. 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!=0`. So a number such as `3` could be written as `3x^0`, if `x!=0`. When `x!=0` it can be written as `3x^0 = 3 * 1 = 3`.)
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Example |
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Problem |
Identify the coefficient, variable, and exponent of the monomial `3/5k^8`. |
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Answer
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The variable is `k`. The exponent of `k` is `8`. The coefficient of `k^8` is `3/5`. |
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Example |
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Problem |
Identify the coefficient, variable, and exponent of `x`. |
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Answer |
The variable is `x`. |
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The exponent of `x` is `1`. |
`x = x^1`, so the exponent is `1`. |
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The coefficient of `x` is `1`. |
`x = 1x^1` so the coefficient is also `1`. |
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Identify the coefficient, variable, and exponent of `3y`.
A) The variable is `y`, the exponent is `3`, and the coefficient is `1`.
B) The variable is `y`, the exponent is `0`, and the coefficient is `3`.
C) The variable is `y`, the exponent is `1`, and the coefficient is `3`.
D) The variable is `y`, the exponent is `3`, and the monomial has no coefficient.
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A polynomial is a monomial or the sum or difference of two or more polynomials. Each monomial is called a term A number or product of a number and variables raised to powers. `4x`, `-5y^2`, `6`, and `x^3y^4` are all examples of terms. of the polynomial.
Some polynomials have specific names indicated by their prefix.
monomial A polynomial with exactly one term. `4x`, `-5y^2`, and `6` are all examples of monomials. - is a polynomial with exactly one term (“mono” - means one)
binomialA polynomial with exactly two terms, such as `5y^2 - 4^x` and `x^5 + 6`. - is a polynomial with exactly two terms (“bi” - means two)
trinomialA polynomial with exactly three terms, such as `5y^2 - 4y + 4` and `x^2 + 2xy +y^2`. - 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 `2x^3 + 3x^2 + 8x + 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.
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Monomials |
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`15` |
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`1/2x` |
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`-4y^3` |
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`16n^4` |
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Binomials |
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`3y + 13` |
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`4p - 7` |
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`3x^2+5/8x` |
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`14y^3 + 3y` |
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Trinomials |
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`x^3 - x^2 + 1` |
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`3x^2 + 2x - 9` |
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`3y^3 + y^2 - 2` |
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`a^7+2a^5-3a^3` |
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Other Polynomials |
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`5x^4 + 3x^3 - 6x^2 + 2x` |
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`1/3x^5-2x^4+2/9x^3-x^2+4x-5/6` |
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`3t^3 - 3t^2 - 3t - 3` |
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`q^7 + 2q^5 - 3q^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, `14y^3 + 3y`, could be written as a trinomial, `14y^3 + 0y^2 + 3y`.
A term without a variable is called a constantA symbol that represents a quantity that cannot change. It can be a number, letter or a symbol. term, and the degree of that term is `0`. For example `13` is the constant term in `3y + 13`. You would usually say that `14y^3 + 3y` has no constant term or that the constant term is `0`.
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Which of the following expressions are polynomials?
`2x^4 - 3x^3`
`14`
`(x+1)/x`
A) Only `2x^4 - 3x^3` and `(x+1)/x` are polynomials.
B) Only `2x^4 - 3x^3` and `14` are polynomials.
C) Only `2x^4 - 3x^3` is a polynomial.
D) None of the expressions is a polynomial.
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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.
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Example |
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Problem |
Evaluate `3x^2 - 2x + 1` for `x = -1`. |
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`3(-1)^2 - 2(-1) + 1` |
Substitute `-1` for each `x` in the polynomial. |
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`3(1) - 2(-1) + 1` |
Following the order of operations, evaluate exponents first. |
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`3 + (-2)(-1) + 1` |
Multiply `3` times `1`, and then multiply `-2` times `-1`. |
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`3 + 2 + 1` |
Change the subtraction to addition of the opposite. |
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Answer |
`3x^2 - 2x + 1 = 6`, for `x = -1` |
Find the sum. |
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Example |
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Problem |
Evaluate `-2/3p^4+2p^3-p " for " p=3` |
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`-2/3(3)^4+2(3)^3-3` |
Substitute `3` for each `p` in the polynomial. |
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`-2/3(81)+2(27)-3` |
Following the order of operations, evaluate exponents first and then multiply. |
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`-54+54-3` |
Add and then subtract to get `-3`. |
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Answer |
`-2/3p^4+2p^3-p=-3`, for `p = 3` |
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Evaluate `3x^3 - 2x^2` for `x = -2`.
A) `-24 - 2x^2`
B) `-32`
C) `-16`
D) `16`
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A polynomial may need to be simplified. One way to simplify a polynomial is to combine the like termsTerms that contain the same variables raised to the same powers. For example, `3x` and `-8x` are like terms, as are `8xy^2` and `0.5xy^2`. 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, `3x^2` and `-5x^2` are like terms: They both have `x` as the variable, and the exponent is `2` for each. However, `3x^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.
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Monomials |
Terms |
Explanation |
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`3x`
`14x` |
like |
same variables with same exponents |
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`16z^2`
`-5z^2` |
like |
same variables with same exponents |
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`3x`
`5y` |
unlike |
different variables (although the same exponents) |
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`-3z`
`-3z^2` |
unlike |
same variables but with different exponents |
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Example |
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Problem |
Which of these terms are like terms? `7x^3, 7x, 7y, -8x^3, 9y, -3x^2, 8y^2` |
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`x`: `7x^3, 7x, -8x^3, -3x^2` `y`: `7y, 9y, 8y^2` |
Like terms must have the same variables, so first identify which terms use the same variables. |
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The `x` terms `7x^3` and `-8x^3` have the same exponent. The `y` terms `7y` and `9y` have the same exponent. |
Like terms must also have the same exponents. Identify which terms with the same variables also use the same exponents. |
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Answer |
`7x^3` and `-8x^3` are like terms. `7y` and `9y` are like terms. |
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Which of these are like terms?
`-3a, 3a^2, 8b, -3b^3, 8a, 14b^2, 9a`
A) Only `-3a`, `8a`, and `9a` are like terms.
B) Only `8a` and `9a` are like terms.
C) `8b`, `-3b^3`, and `14b^2` are like terms, and `-3a`, `3a^2`, `8a`, and `9a` are like terms.
D) `3a^2` and `14b^2` are like terms, and `-3a`, `8b`, `8a`, and `9a` are like terms.
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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.
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Example |
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Problem |
Simplify `3x^2 - 5x^2`. |
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`3(x^2) - 5(x^2)` |
`3x^2` and `5x^2` are like terms. |
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`(3 - 5)(x^2)` |
We can rewrite the expression as the product of the difference. |
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`(-2)(x^2)` |
Calculate `3 - 5`. |
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Answer |
`3x^2 - 5x^2 = -2x^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.
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Example |
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Problem |
Simplify `6a^4 + 4a^4`. |
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`6a^4 + 4a^4` |
Notice that both terms have a number multiplied by `a^4`. This makes them like terms. |
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`(6 + 4)(a^4)` |
Combine the coefficients, `6` and `4`. |
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`(10)(a^4)` |
Calculate the sum. |
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Answer |
`6a^4 + 4a^4 = 10a^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.
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Example |
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Problem |
Simplify `3x^2 + 3x + x + 1 + 5x`. |
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`3x`, `x`, and `5x` are like terms |
First identify which terms are like terms: only `3x`, `x`, and `5x` are like terms. |
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`3x^2 + 3x + x + 1 + 5x=3x^2+(3x+x+5x)+1` |
Use the commutative and associative properties to group the like terms together. |
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`3x^2 + (3 + 1 + 5)x + 1` |
Add the coefficients of the like terms. Remember that the coefficient of `x` is `1` because `x=1x`. |
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`3x^2 + (9)x + 1` |
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Answer |
`3x^2 + 3x + x + 1 + 5x=3x^2+9x+1` |
Write the sum as the new coefficient. |
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Simplify by combining like terms. `-3a + 3a^2 + 8a + 9a - 3`
A) `3a^2 + 17a - 3`
B) `3a^2 + 14a - 3`
C) `17a - 3`
D) `17a - 3a - 3`
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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.