What polynomial identity should be used to prove that 162 = (10 6)2?
Learning Objectives
- Identify the terms, the coefficients, and the exponents of a polynomial
- Evaluate a polynomial for given values of the variable
- Simplify polynomials by collecting like terms
Identify the terms, the coefficients, and the exponents of a polynomial
Polynomials are algebraic expressions that are created by combining numbers and variables using arithmetic operations such every bit add-on, subtraction, multiplication, sectionalisation, and exponentiation. You tin create a polynomial by adding or subtracting terms. Polynomials are very useful in applications from scientific discipline and engineering to business organization. You may see a resemblance between expressions, which nosotros have been studying in this course, and polynomials. Polynomials are a special sub-group of mathematical expressions and equations.
The following tabular array is intended to help you lot tell the deviation between what is a polynomial and what is non.
| IS a Polynomial | Is NOT a Polynomial | Because |
| [latex]2x^two-\frac{1}{ii}10 -9[/latex] | [latex]\frac{2}{x^{2}}+x[/latex] | Polynomials but have variables in the numerator |
| [latex]\frac{y}{4}-y^iii[/latex] | [latex]\frac{2}{y}+4[/latex] | Polynomials merely accept variables in the numerator |
| [latex]\sqrt{12}\left(a\correct)+9[/latex] | [latex]\sqrt{a}+seven[/latex] | Variables nether a root are not immune in polynomials |
The bones building block of a polynomial is a monomial. A monomial is one term and tin 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.
Examples of monomials:
- number: [latex]{2}[/latex]
- variable: [latex]{x}[/latex]
- product of number and variable: [latex]{2x}[/latex]
- product of number and variable with an exponent: [latex]{2x}^{three}[/latex]
The coefficient tin can be any real number, including 0. The exponent of the variable must be a whole number—0, 1, 2, 3, and then on. A monomial cannot accept 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 i. And a monomial with no variable has a degree of 0. (Since [latex]ten^{0}[/latex] has the value of ane if [latex]10\neq0[/latex], a number such as 3 could likewise be written [latex]3x^{0}[/latex], if [latex]x\neq0[/latex] as [latex]3x^{0}=3\cdot1=3[/latex].)
Example
Identify the coefficient, variable, and degree of the variable for the post-obit monomial terms:
1) 9
2) x
3) [latex] \displaystyle \frac{3}{five}{{yard}^{8}}[/latex]
A polynomial is a monomial or the sum or difference of ii or more than polynomials. Each monomial is chosen a term of the polynomial.
Some polynomials take specific names indicated by their prefix.
- monomial—is a polynomial with exactly 1 term ("mono"—means one)
- binomial—is a polynomial with exactly two terms ("bi"—means two)
- trinomial—is a polynomial with exactly iii terms ("tri"—ways 3)
The word "polynomial" has the prefix, "poly," which means many. Nevertheless, the word polynomial can exist used for all numbers of terms, including just 1 term.
Because the exponent of the variable must be a whole number, monomials and polynomials cannot have a variable in the denominator.
Polynomials tin exist classified by the degree of the polynomial. The degree of a polynomial is the caste of its highest degree term. So the degree of [latex]2x^{3}+3x^{2}+8x+v[/latex] is 3.
A polynomial is said to exist written in standard form when the terms are arranged from the highest degree to the lowest degree. When information technology is written in standard form information technology is easy to determine the caste of the polynomial.
The table below illustrates some examples of monomials, binomials, trinomials, and other polynomials. They are all written in standard form.
| Monomials | Binomials | Trinomials | Other Polynomials |
| xv | [latex]3y+xiii[/latex] | [latex]x^{iii}-x^{2}+1[/latex] | [latex]5x^{4}+3x^{three}-6x^{2}+2x[/latex] |
| [latex] \displaystyle \frac{i}{2}10[/latex] | [latex]4p-7[/latex] | [latex]3x^{2}+2x-ix[/latex] | [latex]\frac{1}{3}x^{5}-2x^{iv}+\frac{two}{9}ten^{three}-x^{2}+4x-\frac{5}{6}[/latex] |
| [latex]-4y^{3}[/latex] | [latex]3x^{2}+\frac{5}{8}10[/latex] | [latex]3y^{3}+y^{2}-2[/latex] | [latex]3t^{3}-3t^{two}-3t-three[/latex] |
| [latex]16n^{4}[/latex] | [latex]14y^{three}+3y[/latex] | [latex]a^{7}+2a^{5}-3a^{three}[/latex] | [latex]q^{7}+2q^{5}-3q^{3}+q[/latex] |
When the coefficient of a polynomial term is 0, you lot commonly practice not write the term at all (because 0 times anything is 0, and adding 0 doesn't change the value). The concluding binomial higher up could be written as a trinomial, [latex]14y^{3}+0y^{2}+3y[/latex].
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 [latex]3y+13[/latex]. Y'all would usually say that [latex]14y^{3}+3y[/latex] has no abiding term or that the constant term is 0.
Case
For the following expressions, determine whether they are a polynomial. If and then, categorize them every bit a monomial, binomial, or trinomial.
- [latex]\frac{x-3}{1-x}+x^ii[/latex]
- [latex]t^two+2t-3[/latex]
- [latex]10^three+\frac{ten}{eight}[/latex]
- [latex]\frac{\sqrt{y}}{2}-y-1[/latex]
In the post-obit video, you will exist shown more examples of how to identify and categorize polynomials.
Evaluate a polynomial for given values of the variable
Yous can evaluate polynomials just as you have been evaluating expressions all along. To evaluate an expression for a value of the variable, yous substitute the value for the variable every time it appears. And so employ the society of operations to find the resulting value for the expression.
Example
Evaluate [latex]3x^{2}-2x+1[/latex] for [latex]x=-i[/latex].
Example
Evaluate [latex] \displaystyle -\frac{2}{3}p^{4}+2p^{iii}-p[/latex] for [latex]p = three[/latex].
The following video presents more than examples of evaluating a polynomial for a given value.
Simplify polynomials by collecting similar terms
Apple tree and Orange
A polynomial may need to be simplified. Ane 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, [latex]3x^{2}[/latex] and [latex]-5x^{2}[/latex] are like terms: They both have x as the variable, and the exponent is 2 for each. However, [latex]3x^{ii}[/latex] and [latex]3x[/latex] are not like terms, considering their exponents are different.
Here are some examples of terms that are alike and some that are unlike.
| Term | Like Terms | UNLike Terms |
| [latex]a[/latex] | [latex]3a, \,\,\,-2a,\,\,\, \frac{i}{2}a[/latex] | [latex]a^2,\,\,\,\frac{1}{a},\,\,\, \sqrt{a}[/latex] |
| [latex]a^2[/latex] | [latex]-5a^ii,\,\,\,\frac{i}{4}a^2,\,\,\, 0.56a^2[/latex] | [latex]\frac{1}{a^ii},\,\,\,\sqrt{a^2},\,\,\, a^3[/latex] |
| [latex]ab[/latex] | [latex]7ab,\,\,\,0.23ab,\,\,\,\frac{2}{iii}ab,\,\,\,-ab[/latex] | [latex]a^2b,\,\,\,\frac{1}{ab},\,\,\,\sqrt{ab} [/latex] |
| [latex]ab^two[/latex] | [latex]4ab^2,\,\,\, \frac{ab^2}{7},\,\,\,0.4ab^2,\,\,\, -a^2b[/latex] | [latex]a^2b,\,\,\, ab,\,\,\,\sqrt{ab^two},\,\,\,\frac{1}{ab^2}[/latex] |
Example
Which of these terms are like terms?
[latex]7x^{3}7x7y-8x^{3}9y-3x^{2}8y^{2}[/latex]
Yous tin can use the distributive property to simplify the sum of like terms. Call up that the distributive property of addition states that the product of a number and a sum (or deviation) is equal to the sum (or difference) of the products.
[latex]2\left(3+6\right)=two\left(3\correct)+2\left(six\correct)[/latex]
Both expressions equal xviii. And so you can write the expression in whichever form is the most useful.
Permit's see how we can use this property to combine like terms.
Example
Simplify [latex]3x^{ii}-5x^{2}[/latex].
You may have noticed that combining similar terms involves combining the coefficients to detect the new coefficient of the like term. You can use this every bit a shortcut.
Example
Simplify [latex]6a^{iv}+4a^{4}[/latex].
When you accept a polynomial with more terms, you have to be careful that you combine but like terms. If two terms are not like terms, you can't combine them.
Instance
Simplify [latex]3x^{2}+3x+ten+1+5x[/latex]
Summary
Polynomials are algebraic expressions that contain whatsoever number of terms combined by using improver or subtraction. A term is a number, a variable, or a production of a number and one or more variables with exponents. Similar terms (same variable or variables raised to the same power) can be combined to simplify a polynomial. The polynomials tin can be evaluated by substituting a given value of the variable into each instance of the variable, so using gild of operations to complete the calculations.
Source: https://courses.lumenlearning.com/suny-beginalgebra/chapter/read-define-and-evaluate-polynomials/
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