CPS- Mathematic Unit-7

Teaching Of Algebra At Upper Primary Level

(Std.VI to VIII)

What Are Single Variable Equations?

To understand a single variable equation, let us break it into its two components: the variable and the equation.

A variable is a symbolic placeholder for a number we do not yet know. It’s very common to see x or y used as a variable in math problems, but variables can be represented by any symbol or letter.

x+4=14

In this case, x is our variable. It represents a number that is currently unknown.

An equation sets two mathematical expressions equal to one another. This equality is represented with an equals sign (=) and each side of the expression can be as simple as a single integer or as complex as an expression with multiple variables, exponents, or anything else.

The above is an example of an equation. Each side of the expression equals the other.

So if we put together our definitions, we know that:

A single variable equation is an equation in which there is only one variable used. (Note: the variable can be used multiple times and/or used on either side of the equation; all that matters is that the variable remains the same.)

6x+3−2x=19

4y−2=y+7

These are all examples of single variable equations. You can see how some expressions used the variable multiple times or used the variable in both expressions (on either side of the equals sign).

No matter how many times the variable is used, these still count as single variable problems because the variable remains constant and there are no other variables.

How to Manipulate a Single Variable Equation

In order to solve a single variable equation, we must isolate our variable on one side of the equation. And the way we do this is by shifting the rest of our terms to the other side of the equation.

In order to shift our terms (numbers), we must therefore cancel them out on their original side by performing the opposite function of the term.

 Opposite function pairs are:

1.      Addition and subtraction

2.    Multiplication and division

So if we have a term on one side that has a plus sign (addition), we must subtract that same amount from both sides.

x+2=6

x+2−2=6−2

x=4

If we have a term that is multiplied, we must divide that same amount from both sides.

3x = 18

x=6

Formulae for Squares :

·        (a + b)² = a² + 2ab + b²

·        (a – b)² = a² – 2ab + b²

·        a² + b² = (a + b)² – 2ab

·        a² – b² = (a – b)(a + b)

·        (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

·        (a – b – c)² = a² + b² + c² – 2ab + 2bc – 2ca

·         

An algebraic expression is a mathematical phrase that contains integral or fractional constants (numbers), variables (alphabets) and algebraic operators (such as addition, subtraction, division, multiplication, etc.) operating on them. Also, these expressions are expressed in the form of term, factor and coefficient.

a + 1, a + b, x² + y, 5x/2, etc. are few examples of the algebraic expressions. The algebraic expressions are readily used as a number of mathematical formulas and find usage in generalizing them.

we will learn about the different elements forming these expressions – terms, its factors and coefficient.

Terms of an Expression

A term is a number, a variable, product of two or more variables or product of a number and a variable. An algebraic expression is formed by a single term or by a group of terms. For example, in the expression 4x + y, the two terms are 4x and y.

It is to be noted here that terms add up to form an expression. Say there is a term 8xy, which is the product of 8, x and y. There is another term -4z, which is the product of -4 and z. On adding them up, 8xy + (-4z), we get 8xy – 4z, which is an algebraic expression.

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.
• 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.
• In the term –x, the coefficient is -1.
• 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.

What is a Coefficient in Maths?

A coefficient is an integer that is written along with a variable or it is multiplied by the variable. For example, in the term 2x, 2 is the co-efficient.

Those variables which do not carry any number along with them have a coefficient 1. For example, the term y has coefficient 1.

In linear algebra, when we write the equations, the coefficients always exist for variables.

Expansion Formulae

list of Algebraic Expansion formulas –

• a² – b² = (a – b)(a + b)
• (a+b)² = a² + 2ab + b²
• a² + b² = (a + b)² – 2ab
• (a – b)² = a² – 2ab + b²
• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
• (a – b – c)² = a² + b² + c² – 2ab + 2bc – 2ca
• (a + b)³ = a³ + 3a²b + 3ab² + b³ ; (a + b)³ = a³ + b³ + 3ab(a + b)
• (a – b)³ = a³ – 3a²b + 3ab² – b³
• a³ – b³ = (a – b)(a² + ab + b²)
• a³ + b³ = (a + b)(a² – ab + b²)
• (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴
• (a – b)⁴ = a⁴ – 4a³b + 6a²b² – 4ab³ + b⁴
• a⁴ – b⁴ = (a – b)(a + b)(a² + b²)
• a⁵ – b⁵ = (a – b)(a⁴ + a³b + a²b² + ab³ + b⁴)
• If n is a natural number aⁿ – bⁿ = (a – b)(a^n-1 + a^n-2b+…+ b^n-2a + b^n-1)
• If n is even (n = 2k), aⁿ + bⁿ = (a – b)(a^n-1 + a^n-2b +…+ b^n-2a + b^n-1)
• If n is odd (n = 2k + 1), aⁿ + bⁿ = (a + b)(a^n-1 – a^n-2b +a^n-3b²…- b^n-2a + b^n-1)
• (a + b + c + …)² = a² + b² + c² + … + 2(ab + ac + bc + ….)
• Laws of Exponents (a^m)(aⁿ) = a^m+n ; (ab)^m = a^mb^m ; (a^m)ⁿ = a^mn

Factorization 

In Mathematics, factorization or factoring is defined as the breaking or decomposition of an entity (for example a number, a matrix, or a polynomial) into a product of another entity, or factors, which when multiplied together give the original number or a matrix, etc. This concept we will learn majorly in our lower secondary classes from 6 to 8.

It is simply the resolution of an integer or polynomial into factors such that when multiplied together they will result in original or initial the integer or polynomial. In the factorisation method, we reduce any algebraic or quadratic equation into its simpler form, where the equations are represented as the product of factors instead of expanding the brackets. The factors of any equation can be an integer, a variable or an algebraic expression itself.

Math Factorization

To the factor, a number means to break it up into numbers that can be multiplied to get the original number. For example,

24 = 4 × 6	4 and 6 are the factors of 24
9 = 3 × 3	3 is the factor of 9

Also, numbers can be factorized into different combinations. There are different ways to find the Factors of a Number.  To find the factors of an integer is an easy method but to find the factors of algebraic equations is not that easy. So let us learn to find the factors of quadratic polynomial.

Factorization in Algebra

The numbers -12, -6, -2, -1, 1, 2, 6, and 12 are all factors of 12 because they divide 12 without a remainder. It is an important process in algebra which is used to simplify expressions, simplify fractions, and solve equations. It is also called as Algebra factorization.

Factorisation Of Algebraic Expression

A number or quantity that when multiplied with another number produces a given number or expression. For example, the factors of 12 are 1, 2, 3, 4, 6 and 12.

12 = 1×12

12 = 2×6

12 = 3×4

Any number can be expressed in the form of its factors as explained shown above.

In terms of its prime factors 12 can be expressed as:

12 = 2 × 3 × 2 × 1

Similarly an algebraic expression can also be expressed in the form of its factors. An algebraic expression consists of variables, constants and operators. An algebraic expression consists of terms separated by addition operation. Consider the following algebraic expression:

3xyz–16x2−yz

This expression consists of 3 terms 3xyz, −16x2 and −yz.

Each term of this algebraic expression can be expressed in the form of its factors as:

3xyz=3.x.y.z, −16x2=−1.2.2.2.2.x.x  and −yz=−1.y.z.

Algebraic Expressions can be factorized using many methods. The most common methods used for factorization of algebraic expressions are:

1.      Factorization using common factors

2.    Factorization by regrouping terms

3.    Factorization using identities

Let us discuss these methods one by one in detail:

1.     Factorization using common factors

In order to factorize an algebraic expression, the highest common factors of the terms of the given algebraic expression are determined and then we group the terms accordingly. In simple terms, the reverse process of expansion of an algebraic expression is its factorization.

To understand this more clearly let us take an example.

Example-  −3y2+18y

Solution- The algebraic expression can be re-written as

−3y2+18y=−3.y.y+3.6.y

⇒−3y2+18y=−3.y(y–6)

Consider the algebraic expression -3y( y – 6), if we expand this we will obtain -3y² + 18y.

Factorization by regrouping terms

In some algebraic expressions, it is not possible that every term has a common factor. For instance, consider the algebraic expression 12a + n -na – 12. The terms of this expression do not have a particular factor in common but the first and last term have a common factor of ‘12’ similarly second and third term has n as a common factor. So the terms can be regrouped as:

⇒12a + n – na – 12= 12a – 12 + n – an

⇒12a – 12 – an + n = 12(a -1) –n(a -1)

After regrouping it can be seen that (a-1) is a common factor in each term,

⇒12a + n -na – 12=(a-1) (12 – n)

Thus by regrouping terms we can factorize algebraic expressions.

Factorizing Expressions using standard identities

An equality relation which holds true for all the values of variables in mathematics is known as an identity. Consider the following identities:

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

(a−b)2=a2+b2−2ab

a2−b2=(a+b)(a−b)

On substituting any value of a and b, both sides of the given equations remain the same. Therefore, these equations are identities

Example: Factorize 9x2+4m2+12mx.

Solution: Observe the given expression carefully. This expression has three terms and all the terms are positive. Moreover, the first and the second term are perfect squares. The expression fits the form (a+b)2=a2+b2+2ab where a = 3x, b = 2m.

9x2+4m2+12mx=(3x)2+(2m)2+2.3x.2m

Therefore, 9x2+4m2+12mx=(3x+2m)2

Thus, the required factorization of 9x²+ 4m² + 12mx is (3x + 2m)² by using standard identities.

Polynomial: A polynomial is one or more terms that contain only non-negative integer exponents, and which are combined with addition and subtraction. Note that the prefix "poly" means "many," so some will define a polynomial to requre at least two (or even three) terms. We do not follow that convention here, but consider any individual term to be a polynomial.

Examples: The following are examples of polynomials.
3, 3 + x, y⁵ - 1, 2x + 3y - 20z, x

Descending Order: A polynomial is written in descending order if its terms are arranged in order from largest degree to smallest degree. Note that this is the standard form for a polynomial; rearranging your polynomials to descending order should be automatic.

Examples: The following are polynomials written first in a random order, and then rearranged to descending order.
x³ + 7 - x⁵ in descending order is: -x⁵ + x³ + 7
xy⁵ - xy + x + x³y⁶ in descending order is: x³y⁶ + xy⁵ - xy + x

Degree of a Polynomial: The degree of a polynomial is the largest degree of any of its individual terms. If the polynomial is written in descending order, that will be the degree of the first term.

Examples: The following are examples of polynomials, with degree stated.
x³ + 2x + 1 has degree 3.
x⁵y + x³y² + xy³ has degree 6.

There are two ways to divide polynomials but we are going to concentrate on the most common method here: The algebraic long method or simply the traditional method of dividing algebraic expression.

Algebraic Long Method

Here are the steps in dividing polynomials using the long method:

1.      Arrange the indices of the polynomial in descending order. Replace the missing term(s) with 0.

2.    Divide the first term of the dividend (the polynomial to be divided) by the first term of the divisor. This gives the first term of the quotient.

3.    Multiply the divisor by the first term of the quotient.

4.    Subtract the product from the dividend then bring down the next term. The difference and the next term will be the new dividend. Note: Remember the rule in subtraction "change the sign of the subtrahend then proceed to addition".

5.     Repeat step 2 – 4 to find the second term of the quotient.

6.    Continue the process until a remainder is obtained. This can be zero or is of lower index than the divisor.

If the divisor is a factor of the dividend, you will obtain a remainder equal to zero. If the divisor is not a factor of the dividend, you will obtain a remainder whose index is lower than the index of the divisor.