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Algebra 1 View details

1. Standard Form of a Polynomial

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eCourses /

Algebra 1 /

Chapter 7

Standard Form of a Polynomial

Polynomials are mathematical expressions that can be simplified into a standard form. This form organizes the terms from the highest degree to the lowest. Each term, known as a monomial, plays a crucial role in determining the polynomial's properties. Understanding the standard form is vital for various applications, including engineering, physics, and economics. It serves as the foundation for more complex mathematical operations and problem-solving. Knowing how to convert a polynomial into its standard form can simplify calculations and provide insights into the behavior of mathematical models.

Lesson Settings & Tools

	12 Theory slides
	9 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Image Credits

Slide of 12

Various events in daily life depend on multiple factors. Consider the

CO_{2}

emission of a car. It depends on many variables, such as the gas used, the engine size, and the number of cylinders. Countless daily events like that can be modeled using a combination of powers of different variables. Making those models requires using algebraic expressions with multiple terms called polynomials.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Variable
Algebraic Expressions
Term
Powers
Exponents
Zero Exponent Property

Explore

Classifying Algebraic Expressions

Take a look at the following single-term algebraic expressions. Can a shared characteristic be recognized which would allow for them to be classified reasonably? Drag the expressions and use the colored areas to visualize any common themes, and identify which expressions could be grouped together.

Interactive applet allowing to drag expressions to classify them

Discussion

Defining Monomials

A monomial is an algebraic expression consisting of only one term. It is a product of powers of variables and a constant called the coefficient.

The monomial 2x^2y^3 with a coefficient of 2, and variables x and y.

A single-term expression is a monomial only if all of its variables have whole numbers — non-negative and integers — as exponents. However, variables with positive exponents in the denominator are excluded because they are equivalent to a power in the numerator with the opposite exponent, according to the Quotient of Powers Property. Consider the following example.

\frac{5 x}{y ^{2}} = 5 x y^{- 2}

In other words, if a variable in the denominator of an expression, it is not a monomial. The following are valid examples of monomials.

Expression	Why It Is a Monomial
$5$	Any constant is a valid monomial. By the Zero Exponent Property, $5 x^{0} = 5 .$
$0$	The coefficient of a monomial can be $0 .$
$- 2 x^{5}$	The coefficient can be negative.
$\frac{x ^{3} y}{5}$	A monomial can have numbers in the denominator.

Although they appear to be monomials at first glance, the single-term expressions in the following table do not satisfy the definition of a monomial.

Expression	Why It Is Not a Monomial
$2 x^{- 1}$	The variables of a monomial cannot have negative integer exponents.
$4 \frac{x ^{3}}{y}$	Monomials cannot have variables in the denominator.
$5 x^{3} y^{\frac{1}{2}}$	The variables of a monomial must only have whole number exponents.

Concept

Degree - Monomial

The degree of a monomial is the sum of the exponents of its variable factors. If a variable has no exponent written in it, it is assumed to be $1 .$ Additionally, all nonzero constants have a degree of $0 .$ The constant $0$ does not have a degree.

Monomial	Degree
$3 x$	$1$
$x^{2}$	$2$
$9 x^{3}$	$3$
$x^{3} y$	$4$
$7$	$0$
$\frac{a ^{3} b ^{4} c ^{5}}{1 3}$	$12$
$0$	undefined

Pop Quiz

Identifying Monomials

The next applet shows different algebraic expressions. Use the new knowledge introduced previously to determine whether or not these expressions are monomials.

Interactive graph showing different algebraic expressions

Pop Quiz

Calculating the Degree of a Monomial

The following applet shows different monomials. Determine the given monomial's degree.

Interactive applet showing different monomials

Discussion

Defining Polynomials

A polynomial is either a single monomial or a sum of them. Each of the monomials that form the polynomial are referred to as a term of the polynomial. This definition implies that, for an expression to be a polynomial, each of its terms must be a valid monomial.

Example Polynomial ✓ 5 x y + 2 x y^{2} + 8 x - 9 Not a Polynomial \times 3 x y + 2 x y^{- 2} + 7 x y^{3} - 1

In the above example, the latter expression is not a polynomial because the term

2 x y^{- 2}

is not a monomial. The variables in a monomial can only have whole numbers as exponents.

Classifying Polynomials by Their Number of Terms

One way that polynomials can be classified is according to the number of terms they have. The following table shows the names used for this classification.

Name	Definition	Example
Monomial	A polynomial with a single term.	$3 x^{2} y^{3}$
Binomial	A polynomial with a exactly two terms.	$5 x y + 3 x^{2} y^{3}$
Trinomial	A polynomial with exactly three terms.	$x^{3} - 9 x + 4$

When there are more than three terms, the name polynomial is commonly used.

Concept

Degree - Polynomial

The degree of a polynomial is defined as the greatest of the degrees of its terms. For example, the following is a polynomial of degree

5 .

Interactive graphs showing the degreee of each monomial of the polynomial 3xy+4xy^4-9xy^2-84

The monomial with the highest degree is

4 x y^{4} .

This means that the degree of the polynomial is

1 + 4 = 5 .

In other cases, if a polynomial consists of only a nonzero constant term, the polynomial has degree

0 .

For the special case of the zero polynomial, the degree is undefined. The following table summarizes this information by providing some specific examples.

Polynomial	Degree
$- 5 x^{2} y + x^{2} y^{4} - 11 x^{2} - 3$	$6$
$x - 11 x^{4} + 8 x^{3}$	$4$
$7$	$0$
$0$	Undefined

Depending on the degree of a polynomial, it can have different characteristics. For example, a polynomial with degree

1

is linear, and a polynomial with degree

2

is quadratic.

Concept

Leading Coefficient

In a polynomial, the coefficient in front of the term with the highest degree is called the leading coefficient.

Pop Quiz

Identifying Polynomials

The following applet alternates between different algebraic expressions. Determine whether or not these expressions are polynomials.

Interactive applet showing different algebraic expressions

Pop Quiz

Identifying a Polynomial's Degree and Leading Coefficient

The following applet alternates between different polynomials. Determine the given polynomial's degree or leading coefficient as indicated.

Interactive applet showing different polynomials

Example

Identifying Polynomials and Their Number of Terms

Izabella is improving her understanding of polynomials in her algebra course. However, she has come across some problems where she needs to identify the number of terms of some algebraic expressions and only some of the expressions are polynomials.

Izabella feels confused. Help her solve these problems.

Hint

For an algebraic expression to be a polynomial each of its terms must be a monomial. Therefore, exponents of the variables involved can only be whole numbers.

Solution

For an algebraic expression to be a polynomial, each term must be a valid monomial. A monomial is a single-term expression that is a product of numbers and variables raised to exponents that are whole numbers. Three of the given expressions do not satisfy this definition. That takes them out of the running as being polynomials.

Expression	Reason
$4 x + x^{\frac{1}{2}}$	It has a non-integer exponent.
$2 x^{2} + 5 + y^{- 2}$	It has a negative exponent.
$\frac{3 x}{2 y} + x^{2} - 5 y^{3} + 9$	It has a variable in the denominator.

Now that Izabella knows which expressions are polynomials and which are not, she can figure out how to identify their number of terms. Recall that, in any algebraic expression, terms are separated by a plus or a minus sign.

Polynomial	Number of terms
$\frac{x}{4} - 3 x^{3}$	$2$
$2 x^{2} + 5 - 7 y$	$3$
$\frac{x y}{8} + 3 x^{3} - 3 y^{3} + 2 x$	$4$
Expression	Number of terms
$4 x + x^{\frac{1}{2}}$	$2$
$2 x^{2} + 5 + y^{- 2}$	$3$
$\frac{3 x}{2 y} + x^{2} - 5 y^{3} + 9$	$4$

Discussion

Simplifying and Writing Polynomials in Standard Form

Concept

Standard Form - Polynomial

A polynomial in one variable is expressed in standard form when the monomials that form it are arranged in decreasing degree order. This form can be represented with the following general expression.

$a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x^{1} + a_{0} x^{0}$

In that general expression,

n

is a whole number and the coefficients

a_{n},

a_{n - 1},

\dots,

a_{2},

a_{1},

a_{0}

are real numbers. The following expression written in standard form shows a polynomial with a degree of

5 .

x^{5} - 12 x^{4} - 2 x^{3} + 8 x^{2} + 9 x + 0

It should be noted that coefficients can be zero. In those cases, the corresponding terms are often omitted, which causes consecutive terms to have exponents that are not consecutive descending numbers. The following example, in standard form, shows a polynomial with a degree of

4 .

3 x^{4} - 2 x^{2} + 6 x + 5 ⇕ 3 x^{4} + 0 \cdot x^{3} - 2 x^{2} + 6 x + 5

It can be seen that the

x^{3} -

term was omitted, at first. However, if chosen to do so, the polynomial can be expressed in standard form with a coefficient of

0

and the corresponding term.

Concept

Like Terms

Two or more terms in an algebraic expression are like terms if they have the same variable(s) with the same exponent(s).

7 x y + 2 x - 3 x y - 2 x^{2} + x + 5 x^{2}

In this expression, there are three sets of like terms — namely,

x y -

terms,

x -

terms, and

x^{2} -

terms.

Note that all constants are like terms. Like terms in an expression can be combined by using addition or subtraction to rewrite the expression in simplest form.

Method

Combining Like Terms

In an expression, groups of like terms can be combined to write an equivalent expression with the least possible number of terms. As an example, consider the following expression.

7 x y + 2 x - 3 x y - 2 x^{2} + x + 5 x^{2}

There are three steps to follow to combine like terms.

Identify Like Terms

Start by identifying the like terms in the expression. These are terms with the same variables raised to the same power. Also, all constants are like terms.

This expression has two

x -

terms, two

x y -

terms, and two

x^{2} -

terms.

Grouping Like Terms

Now that the like terms are identified, use the Commutative Property of Addition to rearrange the expression so that the like terms are grouped together.

Add and Subtract the Coefficients

Finally, combine the like terms by adding or subtracting the coefficients of the variables as well as adding or subtracting the constants.

Now that all like terms have been combined, the initial expression has been reduced to its simplest form.

7 x y + 2 x - 3 x y - 2 x^{2} + x + 5 x^{2} ⇕ 4 x y + 3 x + 3 x^{2}

Example

Simplifying Polynomials and Identifying Their Characteristics

Izabella, so excited about solving polynomial related problems, has started a study group with Magdalena to work together on another set of polynomials. This time, the polynomials need to be written in standard form and have their characteristics identified.

Izabella and Magdalena are studying together

Once again, they have run into a problem. The polynomials are not simplified yet. Join their study group and give them a hand.

I . x^{2} + 4 x^{3} - 5 x^{3} + 6 x^{2} + 2 x^{3} II . 8 x + 4 x^{3} - 2 x^{5} + 7 x - 12 x^{3}

a Simplify Polynomial I, then write it in standard form. What type of polynomial is this, according to its number of terms? Lastly, find its degree and leading coefficient.

b Simplify Polynomial II, then write it in standard form. What type of polynomial is this according to its number of terms? Lastly, find its degree and leading coefficient.

Answer

a Standard Form:

x^{3} + 7 x^{2}

Classification: Binomial
Degree: $3$

Leading Coefficient:

1

b Standard Form:

- 2 x^{5} - 8 x^{3} + 15 x

Classification: Trinomial
Degree: $5$

Leading Coefficient:

- 2

Hint

a A polynomial is in standard form when all of its terms are sorted by their degree in descending order.

b A polynomial of exactly one term is called a monomial. If it has two terms, then it is called a binomial and, if it has three, it is called a trinomial.

Solution

a To simplify the polynomial, like terms will be identified and grouped.

x^{2} + 4 x^{3} - 5 x^{3} + 6 x^{2} + 2 x^{3} ⇕ x^{2} + 6 x^{2} + 4 x^{3} - 5 x^{3} + 2 x^{3}

Now, like terms will be combined by adding or subtracting their coefficients.

x^{2} + 6 x^{2} + 4 x^{3} - 5 x^{3} + 2 x^{3}

FactorOut

Factor out $x^{2} & x^{3}$

(1 + 6) x^{2} + (4 - 5 + 2) x^{3}

AddSubTerms

Add and subtract terms

7 x^{2} + x^{3}

Note that this polynomial has exactly two terms. Therefore, it is a binomial. Next, each term will be analyzed to determine its degree and coefficient.

Term	Degree	Coefficient
$7 x^{2}$	$2$	$7$
$x^{3} \Leftrightarrow (1) x^{3}$	$3$	$1$

To write the polynomial in standard form, its terms will be arranged according to their degree, in descending order.

Standard Form x^{3} + 7 x^{2}

Finally, recall that the degree and leading coefficient of a polynomial are the same as those of the monomial of highest degree. Since the highest degree monomial is

(1) x^{3},

the polynomial's degree is

3

and its leading coefficient is

1 .

The following table summarizes all the information found.

Standard Form
$x^{3} + 7 x^{2}$
Classification	Degree	Leading Coefficient
Binomial	$3$	$1$

b The same steps as in Part A will be followed. The polynomial will be simplified by identifying and combining like terms.

8 x + 4 x^{3} - 2 x^{5} + 7 x - 12 x^{3}

CommutativePropAdd

Commutative Property of Addition

8 x + 7 x + 4 x^{3} - 12 x^{3} - 2 x^{5}

FactorOut

Factor out $x & x^{3}$

(8 + 7) x + (4 - 12) x^{3} - 2 x^{5}

AddSubTerms

Add and subtract terms

15 x - 8 x^{3} - 2 x^{5}

Note that this polynomial has exactly three terms. Therefore, it is a trinomial. Now, each term's degree and coefficient will be identified.

Term	Degree	Coefficient
$15 x \Leftrightarrow 15 x^{1}$	$1$	$15$
$- 8 x^{3}$	$3$	$- 8$
$- 2 x^{5}$	$5$	$- 2$

Next, the polynomial will be written in standard form sorting its terms by degree in descending order.

Standard Form - 2 x^{5} - 8 x^{3} + 15 x

Finally, since the highest degree monomial is

- 2 x^{5},

the polynomial's degree is

5

and its leading coefficient is

- 2 .

The information found will be summarized using a table.

Standard Form
$- 2 x^{5} - 8 x^{3} + 15 x$
Classification	Degree	Leading Coefficient
Trinomial	$5$	$- 2$

Closure

Introduction to Polynomial Equations

Now that polynomials have been introduced, equations containing them can be managed. The following example shows a polynomial equation, also known as an algebraic equation.

2 x^{3} - 12 = - 8 x^{2} - 2 x

It is always possible to rearrange these types of equations so that the right-hand side is

0 .

2 x^{3} + 8 x^{2} + 2 x - 12 = 0

In this format, the degree of the polynomial in the left-hand side defines the degree of the equation. It is a remarkable fact that the number of solutions of a polynomial equation is equal to its degree. This relationships is known as the Fundamental Theorem of Algebra. Since the example polynomial equation above is of degree

3,

the associated equation has

3

solutions.

Polynomial Equation	Degree
$2 x^{3} + 8 x^{2} + 2 x - 12 = 0$	$3$
Solutions
$x = - 3, x = - 2, x = 1$

Polynomial equations can be solved algebraically by using methods such as the Quadratic Formula or others based on the Zero Product Property. Alternatively, they can also be solved graphically or by using numerical methods.

Level 1

Level 2

Level 3

Standard Form of a Polynomial

Exercises

Consider the following polynomial.

6 x^{2} + 2 x^{4} - x

What is the expression's first, second, and third term when the polynomial is written in standard form?

Identify the polynomial's degree.

Identify the polynomial's leading coefficient.

The standard form of a polynomial arranges its terms by degree in descending numerical order. Therefore, let's start by arranging the terms accordingly. 6x^2+2x^4-x ⇕ 2x^4 + 6x^2 - x Once the polynomial is in standard form, we can see that 2x^4 is the first term, 6x^2 is the second term, and - x is the third term.

The degree of a polynomial the greatest of the degrees of its terms. Let's begin by highlighting the degrees of each term. 2x^4 + 6x^2 - x ⇕ 2x^4 + 6x^2 - x^1 The terms of the given polynomial have degrees 4, 2, and 1, respectively. Therefore, the degree of the polynomial is 4.

The coefficient that stands in front of the term with the highest degree is the leading coefficient of a polynomial. Alternatively, when written in standard form, the leading coefficient of a polynomial is the constant in front of the first term. 2x^4 + 6x^2 - x In Part B we identified 2x^4 as the term with the highest degree. Therefore, 2 is the leading coefficient of the polynomial.

To calculate the volume of a sphere, we use the formula $V = \frac{4}{3} π r^{3} .$ What is the degree of this monomial?

Let's begin by carefully considering the given formula. V=4/3 π r^3 The degree of a monomial is the sum of the exponents of the variables. Remember that π is not a variable. In fact, it represents an irrational number. π = 3.141592... This means that the left two factors of the formula are constants. V= 4/3 π* r^3 Since r is the only variable in the formula, and it is raised to the third power, we can conclude that the degree of the monomial is 3.

Is the following expression a polynomial? Explain your reasoning.

9 x^{5} + x^{2} - 6.5 x^{4} - 6

3^{x} + 10

9 + x - 7

x^{2} + \frac{1}{x ^{2} + 5}

Polynomials are algebraic expressions that can be written as a sum of terms of the following form. any number* x^(whole number) Let's rewrite some of the variable terms and also the constant term. Then, we can check if it follows the description of a polynomial. any number* x^(whole number) ⇓ 9x^5+ 1x^2 - 6.5x^4 - 6x^0 The expression is a polynomial since all terms follow the description of a polynomial.

Looking at the expression, we can see that one of the terms does not match the description of a term contained within a polynomial. any number* x^(whole number) ⇓ 3^x+ 10x^0 As we can see, x is in the exponent of one of the terms, not the base of the exponent. Therefore, this expression is not a polynomial.

Let's first simplify the expression by combining like terms.

The square root of a number can be rewritten as the number raised to the power of 0.5. If we rewrite sqrt(x) in this way, we notice that the expression is not a polynomial. any number* x^(whole number) ⇓ 2x^0+ 1x^(0.5) Since 0.5 is not a whole number, the expression is not a polynomial.

Any expression written in the form 1a^b can be rewritten as a^(- b). Let's rewrite the expression according to this and determine if it is a polynomial. any number* x^(whole number) ⇓ p(x)= 1x^2+ 1(x^2+5)^(- 1) The expression is not a polynomial because - 1 is not a whole number.

What is the degree of the given monomial?

4 x

23 x^{4}

8 m^{2} n^{4}

5 q^{4} r s^{6}

A monomial is defined as a real number, a variable, or a product of a real number and one or more variables where the exponents are whole numbers. The degree of a monomial is the sum of the exponents of its variables. Let's identify all the exponents. 4x ⇔ 4x^1 As we can see, the exponent of the sole variable x is 1. Therefore, the degree of the monomial is 1.

As we did in Part A, let's identify the exponent of the variable of the monomial. 23x^4 We see that the degree of the monomial is 4.

This time the monomial has two variables. Let's identify their exponents. 8m^2n^4 The degrees of m and n are 2 and 4, respectively. We can find the degree of the monomial by adding these exponents. 2+ 4= 6 Therefore, the degree of the monomial is 6.

Again, lets identify the variables in the monomial. 5q^4rs^6 ⇒ 5q^4 r^1 s^6 The exponents of the variables are 4, 1, and 6. Let's add them to find its degree. 4 + 1 + 6 = 11 The degree of the monomial is 11.

Simplify the following polynomial expressions.

(5 y + 4) + (- 2 y + 6)

(- 3 p^{3} + 5 p^{2} - p) + (15 p - p^{2} - p^{3})

(k^{3} - 7 k + 2) - (k^{2} - 12)

When we simplify an algebraic expression, the first step is to identify which terms can be combined. Remember, only like terms — constant terms or terms with the same variable — can be combined. ( 5y+ 4)+( - 2y+ 6) In the given expression, we have two y-terms and two constants. To simplify the expression, we will rearrange it using the Commutative Property of Addition and then combine like terms.

Let's start by identifying the like terms. ( -3p^3+ 5p^2 - p)+( 15p - p^2 - p^3) Now we can simplify the expression by rearranging the terms and combining them.

As in previous parts, we will start by identifying which terms are like terms. (k^3-7k+ 2)-(k^2 - 12) We only have two like terms, the constants. Let's simplify the expression. Remember that when we remove a parenthesis with a negative sign in front of it, we must change signs of all terms inside the parenthesis.

Mark found a polynomial in one of his old textbooks.

10 x^{8} + x^{4} + 5 x - 14

His niece Maya has a math test on polynomials tomorrow. He decides to quiz her by asking what the degree of the polynomial would be if he...

...increased the second term's coefficient by $3 .$

...decreased the first term's exponent by $4 .$

...increased the third term's exponent by $8 .$

...decreased the constant by $1 .$

The coefficient is the constant that stands in front of a variable term. If we increase the coefficient in front of the second term by 3, we get a new polynomial.

Original Polynomial	New Polynomial
10x^8+x^4+5x-14	10x^8+( 3+1)x^4+5x-14 ⇓ 10x^8+4x^4+5x-14

The polynomial's degree is the greatest of the degrees of its terms, so changing a constant should not have an effect on the degree of the polynomial. We see that 10x^8 is the variable term with the greatest exponent. Therefore, the degree of the new polynomial is 8 — the same as the original polynomial.

If we decrease the first term's exponent by 4, we get the following new polynomial.

Original Polynomial	New Polynomial
10x^8+x^4+5x-14	10x^(8 - 4)+x^4+5x-14 ⇓ 10x^4+x^4+5x-14 ⇓ 11x^4+5x-14

The greatest exponent in the polynomial is now 4, which means that the degree of the new polynomial is 4.

The third term is 5x. Notice that we can write this as 5x^1. Let's increase its exponent by 8.

Original Polynomial	New Polynomial
10x^8+x^4+5x-14	10x^8+x^4+5x^(1 + 8)-14 ⇓ 10x^8+x^4+5x^9-14

Now 5x^9 has the greatest exponent, 9. Therefore, the new polynomial is a ninth-degree polynomial.

We can write the following new polynomial by decreasing the constant by 1.

Original Polynomial	New Polynomial
10x^8+x^4+5x-14	10x^8+x^4+5x-14 - 1 ⇓ 10x^8+x^4+5x-15

The constant will not change the degree of the polynomial. Therefore, the polynomial is still of degree 8.

Standard Form of a Polynomial

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