MONOMIALSA monomial is an algebraic expression that consists of only one term. (A term is a numerical or literal expression with its own sign.) For instance, 9 x, 4 a2, and 3 mpx2 are all monomials. The number in front of the variable is called the numerical coefficient. In 9 x, 9 is the coefficient.

Adding and subtracting monomials

To add or subtract monomials, follow the same rules as with signed numbers,provided that the terms are alike.Notice that you add or subtract the coefficients only and leave the variables the same.

Example 1

Perform the operation indicated.

Remember that the rules for signed numbers apply to monomials as well.

Multiplying monomials

Reminder: The rules and definitions for powers and exponents also apply in algebra.

Similarly, a · a · a · b · b = a3 b2.

To multiply monomials, add the exponents of the same bases.

Example 2

Multiply the following.

1. ( x3)( x4) = x3 + 4 = x7

2. ( x2 y)( x3 y2) = ( x2 x3)( yy2) = x2 + 3 y1 + 2 = x5 y3

3. (6 k5)(5 k2) = (6 × 5)( k5 k2) = 30 k5 + 2 = 30 k7 (multiply numbers)

4. –4( m2 n)(–3 m4 n3) = [(–4)(–3)]( m2 m4)( nn3) = 12 m2 + 4 n1 + 3 = 12 m6 n4 (multiply numbers)

5. ( c2)( c3)( c4) = c2 + 3 + 4 = c9

6. (3 a2 b3 c)( b2 c2 d) = 3( a2)( b3 b2)( cc2)( d) = 3 a2 b3 + 2 c1 + 2 d= 3 a2 b5 c3 d

Note that in example (d) the product of –4 and –3 is +12, the product of m2 and m4 is m6, and the product of n and n3 is n4, because any monomial having no exponent indicated is assumed to have an exponent of l.

When monomials are being raised to a power, the answer is obtained by multiplying the exponents of each part of the monomial by the power to which it is being raised.

Example 3

Simplify.

1. ( a7)3 = a21

2. ( x3 y2)4 = x12 y8

3. (2 x2 y3)3 = (2)3 x6 y9 = 8 x6 y9

Dividing monomials

To divide monomials, subtract the exponent of the divisor from the exponent of the dividend of the same base.

Example 4

Divide.

POLYNOMIALS

A polynomial consists of two or more terms. For example, x + y,y2 – x2, and x2 + 3 x + 5 y2 are all polynomials. A binomial is a polynomial that consists of exactly two terms. For example, x + yis a binomial. A trinomial is a polynomial that consists of exactly three terms. For example, y2 + 9 y + 8 is a trinomial.

Polynomials usually are arranged in one of two ways.Ascending order is basically when the power of a term increases for each succeeding term. For example, x + x2 + x3or 5 x + 2 x2 – 3 x3 +x5 are arranged in ascending order. Descending order is basically when the power of a term decreases for each succeeding term. For example, x3 + x2+ x or 2 x4 + 3 x2 + 7 x are arranged in descending order. Descending order is more commonly used.

Adding and subtracting polynomials

To add or subtract polynomials, just arrange like terms in columns and then add or subtract. (Or simply add or subtract like terms when rearrangement is not necessary.)

Example 1

Do the indicated arithmetic.

Multiplying polynomials

To multiply polynomials, multiply each term in one polynomial by each term in the other polynomial. Then simplify if necessary.

Example 2

Multiply.

Or you may want to use the “ F.O.I.L.” method with binomials.F.O.I.L. means First terms, Outside terms, Inside terms, Last terms. Then simplify if necessary.

Example 3

Multiply.

(3 x + a)(2 x – 2 a) =

Multiply first terms from each quantity.

Now outside terms.

Now inside terms.

Finally last terms.

Now simplify.

6 x2 – 6 ax + 2 ax – 2 a2 = 6 x2 – 4 ax – 2 a2

Example 4

Multiply.

This operation also can be done using the distributive property.

Dividing polynomials by monomials

To divide a polynomial by a monomial, just divide each term in the polynomial by the monomial.

Example 5

Divide.

Dividing polynomials by polynomials

To divide a polynomial by a polynomial, make sure both are in descending order; then use long division. ( Remember: Divide by the first term, multiply, subtract, bring down.)

Example 6

Divide 4 a2 + 18 a + 8 by a + 4. Example 7

Divide.

1.

2.

First change to descending order: x2 + 2 x + 1. Then divide.

3.

Note: When terms are missing, be sure to leave proper room between terms.

4.

5.

This answer can be rewritten as 

BINOMIALBinomial is a little term for a unique mathematical

expression. Learn what makes binomials so special,

how to easily identify them and the mathematical

operations that can be performed on them.

What is a Binomial?

A binomial is a mathematical expression with two terms. Below are some examples of binomials.

Examples of binomials.

All of the above examples are binomials. Look at the above, study them for a bit, and see if you can spot a pattern. The following is a list of what binomials must have.

Must have two terms.

If the variables are the same, then the

exponents must be different.

Exponents must be whole positive integers.

They cannot be negatives or fractions.

A term is a combination of numbers and variables. In the first example, 3x+5, our first term is 3x and our second term is 5. Terms are separated by either addition or subtraction. In our first example, notice how the 3x and 5 are separated by addition. In our last example above, we have a binomial whose two terms both have the same variable s. Notice how each term has its variable to a different exponent. The first term has an exponent of 5 and the second term has an exponent of 4. While we can have fractions for our numbers, we cannot have fractional exponents.

Here is a list showing expressions that are not binomials.

Examples of expressions that are not binomials

Looking at the above, we can say that there are things that binomials cannot have.

- Exponents cannot be negatives or fractions.

- Variables cannot be in the denominator.

Can You Add Binomials?

Now that we can identify binomials, let's see about adding two binomials together. Adding them is fairly straightforward as long as you remember to combine like terms. The caveat here is that many times when you add binomials, your answer won't be a binomial. The only time you will get a binomial back as an answer is if both of your binomials share like terms like in this example.

Adding binomials that share like terms.

Our first binomial is 5x+3y and our second binomial is 4x+7y. The first term of both binomials have the same variable to the same exponent, x. The second term of both binomials also shares a variable to the same exponent, y. We can go ahead and combine the first terms (5x and 4x) together because they are like terms. We can do the same for the second terms (3y and 7y). To combine like terms, we perform the addition or subtraction to the numbers and maintain the variable to its exponent. So 5x+4x=9x and 3y+7y=10y.

Most times, though, you will most likely be adding binomials that don't share like terms.

Adding two random binomials.

In this example, we end up with an expression that is not a binomial. Why? Because the first term in our first binomial has an exponent of 2 but the first term of our second binomial has an exponent of 1. These cannot be combined because they are not like terms. Hence, we have to keep them separated. The second terms of both binomials are like terms and we can combine those (3+7=10).

This is the last type of multiplication that we are going to learn in this unit. The good news is that there is nothing new to learn here. This is just applying the distributive property twice!

The most important part of multiplying two binomials is to make sure that you multiply each term in the first factor by each term in the second. This can get a bit confusing, so be careful!

Let's make things easier by changing subtraction symbol to adding a negative number.

If we apply the distributive property twice, it would look like this:

Binomial

The algebraic expression which has only two terms, then the expression is said to be a binomial. Binomial can be expressed by the operation such as addition, subtraction.

When the polynomials have two term, then the polynomials are called as binomials. For example, in a + b, there are two term a and b separated by addition operation.

BINOMIAL DEFINITION

A polynomial is a finite sum of the terms. A polynomial with two terms is called a binomial.

For example, x + 1, y2 - 3, a2 - b2 are binomials.

A polynomial having two terms is called a binomial.

BINOMIAL IDENTITIES

Binomial identities are given below:

Squaring Binomials

1. (a + b)2 = a2 + b2 + 2ab2. (a - b)2 = a2 + b2 - 2ab3. a2 - b2 = (a - b)(a + b)

Cubing a Binomial

1. (a + b)3 = a3 + b3 + 3a2b + 3ab2

2. (a - b)3 = a3 - b3 - 3a2b + 3ab2

3. a3 - b3 = (a - b)(a2 + ab + b2)4. a3 + b3 = (a + b)(a2 - ab + b2)

FACTORING BINOMIALS

Factoring is a process of writing a polynomial as a product. Binomial is the expression which is a combination of two terms. Let us see with the help of examples how to factor the binomials.

Factoring Binomials Examples

Let us find the factors of 24x + 3x2.

Given 24x + 3x2 

24x = 2 * 2 * 2 * 3 * x

3x2 = 3 * x * x

Take the common factors out

=> 24x + x2 = 2 * 2 * 2 * 3 * x + 3 * x * x

= 3x(8 + x).

MULTIPLYING BINOMIALS

To multiply two binomials, we use the distributive property more than once. When we multiply two binomial, we will end up with 4 terms. Method of multiplying two binomial is also called FOIL method. The most important part of multiplying two binomials is to make sure that you multiply each term in the first factor by each term in the second.

Let us multiply (x + 3)(x - 1)

Given (x + 3)(x - 1)

(x + 3)(x - 1) = x(x - 1) + 3(x - 1)

= x2 - x + 3x - 3

= x2 + 2x - 3.

Dividing BinomialsBack to Top

To divide binomial by binomial or by monomial, use the reverse form of the rule for adding the fractions with a common denominator. We can divide a binomial with a monomial by dividing each term of the binomial by the monomial.

Let us solve x3+8x2+4−2x.

Given x3+8x2+4−2x

x3+8x2+4−2x = (x+2)(x2+4−2x)x2+4−2x

Using the identity a3 + b3 = (a + b)(a2 + b2 - ab), we get

=> x3+8x2+4−2x = x + 2.

Adding BinomialsBack to Top

When terms contain the same variable and same exponent, they are like terms. Addition of binomials is done by combining the like terms.

Let us add 12x4y + 10x4y

Given 12x4y + 10x4y

12x4y + 10x4y = (12 + 10)x4y

= 22 x4y

=> 12x4y + 10x4y = 22 x4y.

SUBTRACTING BINOMIALS

To subtract binomials, follow the same rules as with numbers, provided that the terms are alike. When terms contain the same variable and same exponent, they are like terms. 

Let us subtract 30xy + 3x2 from 10 x2 - 10xy

Given Two binomials 30xy + 3x2 and 10 x2 - 10xy

=> 10 x2 - 10xy - (30xy + 3x2) = 10 x2 - 10xy - 30xy - 3x2 

= (10 - 3)x2 + (-10 - 30)xy

= 7x2 - 40 xy

BINOMIAL RADICAL EXPRESSIONS

A radical expression is an expression containing a square root. Binomial is an expression with two terms, so binomial radical expression consists one of the term in radical form. 

Let us solve 527−−√ + 103√ - 212−−√

Given 527−−√ + 103√ - 212−−√

527−−√ + 103√ - 212−−√ = 532∗3−−−−−√ + 103√ - 222∗3−−−−−√

= 5 * 3 3√ + 103√ - 2 * 2 3√

= (15 + 10 - 4)3√

= 21 3√

SOLVING   BINOMIALS

Given below are some of the examples in solving binomials.

Solved Examples

Question 1: Which of the following referred as binomial?

3x + y

3 * y

3xy

3x ÷ y

Solution:

Option a: 3x + y, addition operation makes the polynomials in two terms.

Option b: 3 * y, multiplication operation makes the polynomials in one term as 3y.

Option c: 3xy, there is one term in this polynomial.

Option d: 3x ÷ y, division operation makes the polynomials in one term.

Hence, the "option a" has the polynomials which have only two term which said to be as binomial.

Question 2: Prove when we multiply binomial with monomial, we get binomial.Solution:

Let take the binomial as (a + b) and monomial as x.

Multiply the binomial with monomial,

(a + b) * x = ax + bx. Thus we get the binomial.

Therefore, we proved as by multiplying binomial and monomials, we get binomial.

Question 3: Factorize 8a3 - y3Solution:

Given 8a3 - y3

8a3 - y3 = 23a3 - y3

= (2a - y)((2a)2 + 2ay + y2)

Using the identity a3 - b3 = (a - b)(a2 + 2ab + b2), we get

=> 8a3 - y3 = (2a - y)(4a2 + 2ay + y2)

Binomial Example

Given below are some of the solved examples on binomials.

Solved Examples

Question 1: What is the area of a square whose side is 2x + 4?

Solution:

Given side of the square = 2x + 4

We know that, the area of the square = side * side

=> Area of the square = (2x + 4)2

Formula for square of binomial:

(a + b)2 = a2 + b2 + 2ab

=> (2x + 4)2 = (2x)2 + 42 + 2(2x) (4)

= 4x2 + 16 + 16x

Question 2: Solve (x + 2)3Solution:

Given (x + 2)3

=> (x + 2)3 = x3 + 23 + 3 * x * 2 * (x + 2)

Using the identity (a + b)3 = a3 + b3 + 3ab(a + b), we get

=> (x + 2)3 = x3 + 8 + 6x(x + 2)

= x3 + 8 + 6x * x + 6x * 2

= x3 + 8 + 6x2 + 12x.

GROUPING SYMBOLSThere are basically three types of grouping symbols: parentheses, brackets, and braces.

Parentheses ( )

Parentheses are used to group numbers or variables. Everything inside parentheses must be done before any other operations.

Example 1

Simplify.

50(2 + 6) = 50(8) = 400

When a parenthesis is preceded by a minus sign, to remove the parentheses, change the sign of each term within the parentheses.

Example 2

Simplify.

Brackets [ ] and braces { }

Brackets and braces also are used to group numbers or variables. Technically, they are used after parentheses. Parentheses are to be used first, then brackets, and then braces: {[( )]}. Sometimes, instead of brackets or braces, you will see the use of larger parentheses.

((3 + 4) · 5) + 2

A number using all three grouping symbols would look like this:

2{1 + [4(2 + 1) + 3]}

Example 3

Simplify 2{1 + [4(2 + 1) + 3]}. Notice that you work from the inside out.

Order of operations

If multiplication, division, powers, addition, parentheses, and so forth are all contained in one problem, the order of operations is as follows:

1. Parentheses

2. Exponents (or radicals)

3. Multiplication or division (in the order it occurs from left to right)

4. Addition or subtraction (in the order it occurs from left to right)

Many students find the made up word PEMDAS helpful as a memory tool. The “P” reminds you that “parentheses” are done first; the “E” reminds you that “exponents” are done next; the “MD” reminds you to “multiply or divide” in the order it occurs from left to right; and the “AS” reminds you to “add or subtract” in the order it occurs from left to right.

Also, some students remember the order using the following phrase:

Please Excuse My Dear Aunt SallyParantheses

Exponents

Multiply or

Divide

Add or

Subtract

Example 4

Simplify the following problems.

Working with negative exponents

Remember, if the exponent is negative, such as x–3, then the variable and exponent may be dropped under the number 1 in a fraction to remove the negative sign as follows.

Example 5

Express the answers with positive exponents.

TERMINOLOGY ON ALGEBRA (1)

Basic algebra terms you need to know are constants, variables, coefficients, terms, expressions, equations and quadratic equations. These are some algebra vocabulary that will be useful.

ConstantsA fixed quantity that does not change. For example: 3,

–6, π, VariablesA variable is a symbol that we assign to an unknown value. It is usually represented by letters such as x,y, or t. For example, we might say that l stands for the length of a rectangle and w stands for the width of the rectangle.We use variables when we need to indicate how objects are related even though we may not know the

exact values of the objects. For example, if we want to say that the length of a rectangle is 3 times the length of its width then we can writel = 3 × w

CoefficientsThe coefficient of a variable is the number that is placed in front of a variable.For example, 3 × w can be written as 3w and 3 is the coefficient.

CoefficientTermsA term can be any of the following:

a constant: e.g. 3, 10, π, 

the product of a number (coefficient) and

a variable: e.g. –3x, 11y, 

the product of two or more variables:

e.g. x2, xy, 2y2, 7xyLike terms are terms that differ only in their numerical

coefficients. For example: 3a, 22a,   are like terms.ExpressionsAn expression is made up of one or more terms.

For example:

3w + 4xy + 5

Equations

An equation consists of two expressions separated by an equal sign. The expression on one side of the equal sign has the same value as the expression on the other side.

For example:

4 + 6 = 5 × 2

l = 3 × w

3w + 4xy + 5 = 2w + 3

Quadratic Equations

A Quadratic Equation is an equation of the form:

ax2 + bx + c = 0, where a, b and c are numbers and a ≠ 0

For example:

x2 + 2x + 3 = 0

2x2 + 5x – 7 = 0

2x2 + 5x = 8     is a quadratic equation because it can be changed to 2x2 + 5x – 8 = 0

x2 + x = 0     is a quadratic with c = 0

2x2 – 7 = 0     is a quadratic with b = 0

2x + 3 = 0     is not a quadratic because a cannot be 0

TERMINOLOGY USED IN ALGEBRA (2)

A variable is also called an unknown and can be represented by letters from the alphabet. Operations in algebra are the same as in arithmetic: addition, subtraction, multiplication and division. An expression is a group of numbers and variables, along with operations. An equation is the equality of two expressions.

VARIABLES AND CONSTANTS

Algebra started off as the addition and subtraction of similar objects. For example, you could add three apples plus two apples to equal five apples. If you substituted the letter "a" for apples, you would have 3 a + 2 a = 5 a in place of 3 apples plus 2 apples equals 5 apples.

Thus, a letter such as "a" could be used for any object. In fact, it could even represent another number or even group of numbers and objects.

Variables

The basis of algebra is that the various mathematical operations can be applied for no matter what you have for "a", "b", or "x".

These letters are called variables, because they can vary and be almost anything. They are also often called unknowns.

Constants

In some situations a quantity may be known, but it is convenient to designate it as a letter. In such a case, the letter represents a constant.

In Einstein's famous equation E = mc2, the letter c represents the speed of light and is thus a constant. It is easier to use c than to write out the speed of light as 186,000 miles per second (or 300,000 km per second).

In some cases, a constant can be an unknown, but it does not necessarily vary.

Common convention

Although any alphabetical letter can be a variable, a common convention used is to designate letters toward

the end of the alphabet to be variables and letters to the front as constants.

Thus, variables are usually represented by the letters x, y or z and constants are a, b or c.

In Einstein's equation, c is a constant, but m is a variable because it represents the unknownmass of an object (thus the reason to use m instead of some other letter).

OPERATIONS

Mathematical operations are addition, subtraction, multiplication and division. Their common symbols are:

Addition + Subtraction − Multiplication × Division ÷

Addition

The following are designated by x + y:

Add x and y

Find the sum of x and y

Increase x by y

Subtraction

The following are all designated by x − y:

Subtract x from y

Find the difference between x and y

x is decreased by y

x take away y

x minus y

x less y

The following are designated by y − x:

x less than y

x from y

Multiplication and division have some variations.

Multiplication sign

Since the letter "x" is often use as a variable or unknown in algebra, this can cause confusion with the multiplication sign ×. Thus, an asterisk (*) or a dot (·) is often used to indicate multiplication instead of ×.

Instead of x × y, multiplication is designated as x*y or x·y.

But another problem then pops up. If you use the dot when dealing with numbers, it may be confused with a decimal point. For example, writing 2 times 5 as 2·5 might be confused with 2½or 2.5. With numbers, it is better to use *, as in 2*5, or even the × sign.

Using no multiplication sign

Mathematicians decided to completely drop the multiplication sign altogether, when multiplying variables or constantsr. Instead of writing x times y times z as x*y*z, it is usually written asxyz in algebra.

Numbers first

When including numbers in the multiplication, the number is written first and no multiplication sign is used. Thus, x times 3 is written as 3x, and x times 2 times a is written as 2ax.

Multiplying numbers together

But if you are multiplying two or more numbers together, you must include a multiplication sign.2 times 5 times 3 is not written as 253 but as 2*5*3.

If there is a letter involved, try to make it as clear as possible. 2*5x or 2*5*x are both acceptable.

Division sign

The slash (/) is also used to denote division. Thus, a ÷ b and a/b both mean "a divided by b."

 a — b  also can denote the division of a by b or a fraction. Since it is difficult to write on a web page, it is seldom used in websites.

Other algebraic operations

Other algebraic operations, such as square root and exponent will be explained in other lessons.

EXPRESSIONS AND EQUATIONS

Expressions lead into equations.

Expressions

An expression is any group or collection of algebraic numbers and variables, including mathematical operations such as addition, division, etc. Examples of expressions include:

3 + 5

2a + 3x − 6/7

5abc

Some expressions may be long and complex, even including parentheses:

3x + (2z − y)/x + 125y − (x + y)/(z +2)

Terms

An expression consists of one or more terms that are separated by an addition or subtraction operation. The expression 2a + 3x − 6/7 consists of the terms (separated by commas):

2a, 3x, − 6/7

Equations

An equation consists of expressions separated by an equal sign. The assumption is that the expressions on the left side of the equal sign are equal to those on the right side.

3 + 5 = 8

5x − 3y = 4z

27 + x = 17/3x

Since some of these equations contain unknowns or variables, they require a solution to verify the equation is valid.

3x − 7 = 2 is valid with when x = 3.

Some equations are equalities when the values are known or the solution is trivial. 3 + 5 = 8and x = 3 are considered equalities.

Summary

Major terms used in algebra include: variables, constants, operations, expressions and equations. A variable can be represented by letters from the alphabet.

While the addition and subtraction operations are the same in algebra as in arithmetic, there are some different designations of multiplication and division used in web pages.

An expression is a group of numbers and variables, along with operations. An equation is the equality of two expressions.