CHAPTER-11 ALGEBRAIC EXPRESSIONS

Learning Outcomes:-

To define Algebraic Expressions,

polynomials, terms, factors,

coefficients, etc.

To add, subtract, multiply and divide

Algebraic Expressions.

To simplify Algebraic Expressions

Algebraic Terms &

Expressions

In mathematics, often the value of a certain

number may be unknown.

A variable is a symbol, usually a letter, which

is used to represent an unknown number.

Some examples of variables are:

x, a, t, y, b

A term can be a number, a variable, or a

number and variable combined by multiplication or division.

Some examples of terms are:

x, 8, 4y,

An expression can be term or a collection of

terms separated by addition or subtraction

operators. Some examples of expressions,

with the numbers of terms, are listed below:

Polynomial

A polynomial looks like this

Polynomial comes from poly-(meaning "many") and -nomial (in this case

meaning "term") ... so it says "many

terms"

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A polynomial can have:

constants (like 3, −20, or ½)

Variables (like x and y)

Exponents (like the 2 in y2), but only 0, 1, 2, 3,

... etc are allowed

that can be combined using addition, subtraction, multiplication and division ...

... except ...

... not division by a variable (so something

like 2/x is right out)

So:

A polynomial can have constants, variables and exponents,

but never division by a variable.

Also they can have one or more terms, but not an infinite number of terms.

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Polynomial or Not?

These are polynomials:

3x x − 2 −6y2 − (79)x

3xyz + 3xy2z − 0.1xz − 200y + 0.5 512v5 + 99w5 5

(Yes, "5" is a polynomial, one term is allowed, and it can be just a constant!)

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These are not polynomials

3xy-2 is not, because the exponent is "-2"

(exponents can only be 0,1,2,...) 2/(x+2) is not, because dividing by a

variable is not allowed 1/x is not either

√x is not, because the exponent is "½"

But these are allowed:

x/2 is allowed, because you can divide by

a constant also 3x/8 for the same reason √2 is allowed, because it is a constant (=

1.4142...etc)

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Monomial, Binomial,

Trinomial

There are special names for polynomials with 1, 2 or 3 terms:

How do you remember the names?

Think cycles!

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There is alsoquadrinomial (4 terms) and quintinomial (5 terms).

Degree of Polynomial

The degree of a polynomial with only one

variable is the largest exponent of that variable.

Example:

The Degree is 3 (the

largest exponent of x)

Youtube Video Links for better understanding

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https://youtu.be/xmJjQ3KyTdw

https://youtu.be/MPcZ3nhZO-M

Factors and coefficients of a Polynomial

FACTOR OF A TERM

The numbers or variables that are multiplied to form a term are called its factors. Example, 5xy is a term with

factors 5, x and y.

The factors cannot be further factorized. Example, 5xy cannot be written as the product of factors 5 and xy. This is

because xy can be factorized to x and y.

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The factors of the term 3a4 are 3, a, a,

a and a.

1 is not taken as a separate factor.

COEFFICIENT OF A TERM

A coefficient is the numerical factor of a term containing constant and variables.

In the term 5ab, 5 is the coefficient.

-5 is the coefficient of the term –5ab2.

When there is no numerical factor in a term, its coefficient is taken as +1. For example, in the term x2y3, the coefficient is

+1.

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A coefficient is sometimes generalized as either the numerical factor, variable factor

or the products of the two. As for example, in the term 5ab2, b is the coefficient of 5ab. Similarly, in 10ab, -2a is the

coefficient of -5b and so on.

Like and Unlike Terms

The terms which have the same literal coefficients raised to the same powers but may only differ in numerical coefficient are

called similar or like terms.

For example:

(i) 3m and –7m are like terms

(ii) z and 3/2 z are like terms

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The terms which do not have the same literal coefficients raised to the same powers are

called dissimilar or unlike terms.

For example:

(i) 9p and 9q are unlike terms

(ii) x/3 and y/3 are unlike terms

YouTube Video Link for better understanding:-https://youtu.be/Jw-

toLAUqPg

Exercise

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Exercise Answer

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Operation on Algebraic

Expression-Addition

In addition of algebraic expressions while

adding algebraic expressions we collect the like terms and add them. The sum of several like terms is the like term whose coefficient is the sum of the coefficients of

these like terms.

Two ways to solve addition of algebraic

expressions.

Horizontal Method

Column Method

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Horizontal Method

Horizontal Method: In this method, all

expressions are written in a horizontal line and then the terms are arranged to collect all

the groups of like terms and then added.

Examples on addition of algebraic expressions:

1. Add: 6a + 8b - 7c, 2b + c - 4a and a - 3b - 2c

Solution:

Horizontal Method: (6a + 8b - 7c) + (2b + c - 4a) + (a - 3b - 2c)

= 6a + 8b - 7c + 2b + c - 4a + a - 3b - 2c Arrange the like terms together, then add.

Thus, the required addition

= 6a - 4a + a + 8b + 2b - 3b - 7c + c - 2c

= 3a + 7b - 8c

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Column Method

Column Method: In this method each expression is written in a separate row such that there like

terms are arranged one below the other in a

column. Then the addition of terms is done

column wise.

Add: 8x² - 5xy + 3y², 2xy - 6y² + 3x² and y² + xy - 6x². Solution:

Arranging the given expressions in descending powers of x with like terms under each other and adding column wise;

8x² - 5xy + 3y² 3x² - 2xy - 6y²

-6x² + xy + y² _____________ 5x² - 2xy - 2y² _____________

= 5x² - 2xy - 2y²

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Exercise

Answers

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Subtraction

Subtraction of algebraic expressions are

explained in each steps:

Steps I: Arrange the terms of the given

expressions in the same order.

Steps II: Write the given expressions in two rows

in such a way that the like terms occur one below

the other, keeping the expression to be

subtracted in the second row.

Steps III: Change the sign of each term in the

lower row from + to - and from - to +.

Steps IV: With new signs of the terms of lower

row, add column wise.

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Ex:- Subtract 3x² - 6x - 4 from 5 + x - 2x².

Solution:

Arranging the terms of the given expressions in

descending powers of x and subtracting column-

wise;

- 2x² + x + 5

+ 3x² - 6x - 4

(-) (+) (+)

_____________

- 5x² + 7x + 9

_____________

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Exercise

Answer

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Multiplication of Two

Monomials

if x is a variable and m, n are positive integers,

then

(xᵐ × xⁿ) = 𝑥𝑚+𝑛

Rule:

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Product of two monomials = (product of their numerical coefficients) × (product of their

variable parts)

Find the product of: (i) 6xy and -3x²y³

Solution:

(6xy) × (-3x²y³)

= {6 × (-3)} × {xy × x²y³}

= -18x³y⁴.

Multiplication of

Polynomials

Suppose (a + b) and (c + d) are two binomials. By

using the distributive law of multiplication over

addition twice, we may find their product as given

below.

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(a + b) × (c + d)

= a × (c + d) + b × (c + d)

= (a × c + a × d) + (b × c + b × d)

= ac + ad + bc + bd

Note: This method is known as the horizontal

method.

https://youtu.be/wUYa2NAV5t4

(i) Multiply (3x + 5y) and (5x - 7y).

Solution:

(3x + 5y) × (5x - 7y)

= 3x × (5x - 7y) + 5y × (5x - 7y)

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= (3x × 5x - 3x × 7y) + (5y × 5x - 5y × 7y)

= (15x² - 21xy) + (25xy - 35y²)

= 15x² - 21xy + 25xy - 35y²

= 15x² + 4xy - 35y².

Column Method of

Multiplication of

Polynomials

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(i) Multiply (5x² – 6x + 9) by (2x -3)

5x² – 6x + 9

× (2x - 3)

____________________

10x³ - 12x² + 18x ⇐ multiplication by 2x.

- 15x² + 18x - 27 ⇐ multiplication by -3.

______________________

10x³ – 27x² + 36x - 27 ⇐ multiplication by (2x - 3). ______________________

Therefore, (5x² – 6x + 9) by (2x - 3) is 10x³ – 27x² + 36x –

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Exercise

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Division of Monomial by

Monomial

In division of algebraic expression if x is a variable and m, n are positive integers such that

m > n then (xᵐ ÷ xⁿ) = 𝑥𝑚−𝑛.

Quotient of two monomials is a monomial which is equal to the quotient of their numerical

coefficients, multiplied by the quotient of their

literal coefficients.

Rule:

Quotient of two monomials = (quotient of their numerical coefficients) x (quotient of their

variables

(i) 8x2y3 by -2xy

Solution:

(i) 8x2y3/-2xy

= -4xy2.

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Division of a Polynomial by

a Monomial

Rule:

For dividing a polynomial by a monomial, divide each

term of the polynomial by the monomial. We divide

each term of the polynomial by the monomial and then

simplify.

Divide:

(i) 6x5 + 18x4 - 3x2 by 3x2 Solution: 6x5 + 18x4 - 3x2 by 3x2

= (6x5 + 18x4 - 3x2) ÷ 3x2

= 6/3 + 18/3 - 3/3

=2x3 + 6x2 - 1.

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Division of a Polynomial by

a Polynomial

We may proceed according to the steps given below: (i) Arrange the terms of the dividend and divisor in

descending order of their degrees. (ii) Divide the first term of the dividend by the first term of the divisor to obtain the first term of the quotient.

(iii) Multiply all the terms of the divisor by the first term of the quotient and subtract the result from the dividend.

(iv) Consider the remainder (if any) as a new dividend and proceed as before. (v) Repeat this process till we obtain a remainder

which is either 0 or a polynomial of degree less than

that of the divisor.

https://youtu.be/FTRDPB1wR5Y

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Divide 12 – 14a² – 13a by (3 + 2a).

Solution:

12 – 14a² – 13a by (3 + 2a).

Write the terms of the polynomial (dividend and divisor both) in decreasing

order of exponents of variables.

So, dividend becomes – 14a² – 13a + 12 and divisor becomes 2a + 3.

Divide the first term of the dividend by the first term of the divisor which gives

first term of the quotient.

Multiply the divisor by the first term of the quotient and subtract the product

from the dividend which gives the remainder.

Now, this remainder is treated as, new dividend but the divisor remains the

same.

Now, we divide the first term of the new dividend by the first term of the

divisor which gives second term of the quotient.

Now, multiply the divisor by the term of the quotient just obtained and

subtracts the product from the dividend.

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Thus, we conclude that divisor and quotient are the factors of dividend if the remainder is zero.

Quotient = -7a + 4, Remainder = 0.

Exercise

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Answers