Basic Algebra and Linear Equation concepts for CAT

Algebra plays a crucial role in the CAT, as an average of 35–40% of the 'Quantitative Aptitude' section's problems are algebra-related, and the questions posed from this section are not particularly calculative and can be rapidly solved using options. According to question papers from previous years, despite being algebraic, the queries in this section are extremely logical.

For this, you must first master the fundamentals, then solve a multitude of problems, so that you are familiar with the variety and diversity of problems.

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Remainder Theorem

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If a polynomial (i.e., a rational integral function) f(x) is divided by (x – a), then the remainder is f(a), which is obtained by substituting a for x in f(x).

1. If a polynomial f(x ) is divided by (x + a ), then remainder is f(−a)
2. If a polynomial f(x ) is divided by (x − a ), then remainder is f(a)
3. If a polynomial f(x) is divided by (ax + b), then the remainder is f(\(-\frac{b}{a}\))

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Factor Theorem

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If f(x) be a polynomial (i.e., rational integral function) in x

and f(a) = 0, then (x − a) is a factor of f (x).

By remainder theorem

if f(x) be divided by (x − a) the

remainder is f(a). But f(a) = 0, i.e., there is no remainder.

∴ f(x) is exactly divisible by (x − a)

Hence, (x − a) is a factor of f(x).

1. If f(x)be a polynomial and f(−a) = 0,then (x+a) is a factor of f(x).
2. If f(x) be a polynomial and f(\(-\frac{b}{a}\)) = 0,then(ax + b)is a factor of f(x).

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HCF and LCM of Polynomials

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1. A polynomial D(x) is a divisor of the polynomial P(x) if it is a factor of P(x). Where Q(x) is another polynomial such that P(x) = D(x)⋅Q(x)
2. HCF/GCD (Greatest Common Divisor): A polynomial h(x) is called the HCF or GCD of two or more given polynomials, if h(x) is a polynomial of highest degree dividing each one of the given polynomials.
3. LCM (Least Common Multiple): A polynomial P(x) is called the LCM of two or more given polynomials, if it is a polynomial of smallest degree which is divided by each one of the given polynomials.

For any two polynomials P(x) and Q(x).We have :

P(x).Q(x) = [HCF of P(x) and Q(x)] × [LCM of P(x) and Q(x)]

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Important Formulae

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1. (a + b)² = a² + b² + 2ab
2. (a − b)² = a² + b² − 2ab
3. (a + b)² = (a − b)² + 4ab
4. (a – b)² = (a + b)² − 4ab
5. (a –  b)² = \(\frac{1}{2}\)[ (a + b)² + (a – b)² ]
6. a² –   b² = (a + b)(a − b)
7. (a + b)³ = a³ + b³ + 3ab(a + b)
8. (a − b)³ = a³ − b³ − 3ab(a − b)
9. a³ + b³ = (a + b)(a² + b² − ab)
10. a³ − b³ = (a − b) (a² +b² + ab)
11. (a + b + c)² = a² + b² + c² + 2(ab + bc + ac)
12. a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ac)
13. if a + b + c = 0,  then a³ + b³ + c³ = 3abc
14. a⁴ + b⁴ + a²b² = (a² + b² + ab)(a² + b² − ab)
15. a⁴ − b⁴ = (a − b)(a + b)(a² + b²)
16. a⁸ − b⁸ = (a − b)(a + b)(a² + b²)(a⁴ + b⁴)

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Basic facts to remember

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To factorize cyclic expressions, one must:

(a) Arrange the words in descending or ascending order by the power of one of the characters (or literals).

(b) combining two or three concepts, you can identify a common factor.

(c) Rewrite the elements of the other factor with the next letter's decreasing or increasing power.

(d) Repeat steps until all variables are identified.

Sigma (Σ) notation

Σx = x + y + z, for letters x, y, z

Σxy = xy + yz + zx

Σx²(y−z) = x²(y−z) + y²(z−x) + z²(x− y)

Some important results

1. Σ(x−y) = 0
2. Σ(x² − y²) = 0
3. Σ(x³ − y³) = 0
4. Σx(y − z) = 0
5. Σx²(y² − z²) = 0
6. Σx²(y − z) = − (x − y)( y − z)(z − x)
7. Σx(y² − z² ) = (x−y)(y−z)(z−x)
8. Σ(x−y)³ = 3(x−y)(y−z)(z−x)

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Linear Equations

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An equation of the form ax + by = c, where a, b and c are real numbers is called a linear equation in two variables x and y.

The graph of a linear equation ax + by = c is a straight line.

The value of the variables that satisfy the equation is called the solution (or solution set) of the equation.

Key Features of Linear Equation with Two Variables:

(i) The graph of an equation of the form x = k (where k is a constant) is a straight line parallel to the y-axis k units away from the y-axis.

(ii) The graph of a y = k equation (where k is a constant) is a straight line parallel to the x-axis k units away from the x-axis.

(iii) The solution to the two equations is found at the site of intersection of the two lines.

(iv) the number of solutions to a single linear equation with two variables is infinite.

Simultaneous Linear Equations

Two or more linear equations in two variables form a system of linear simultaneous equations.

e.g., a₁ x + b₁ y = c₁ and a₂ x + b₂ y = c₁

Consistent System: A system is said to be consistent if it has at least one answer and is made up of two simultaneous linear equations.

Inconsistent System: A system with two simultaneous linear equations that does not have a solution is said to be inconsistent.

If two equations, a₁ x + b₁ y = c₁ and a₂ x + b₂ y = c₂, are true at the same time, then

 \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\) \(\implies\) Unique Solution (Independent Equation)

 \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\) \(\implies\) Infinite solution (Dependent Equation)

 \(\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\) \(\implies\) No Solution (Inconsistent Equation)

Properties of graphs of a₁ x + b₁ y = c₁ and a₂ x + b₂ y = c₂

• intersecting if the system has a unique solution.
• coincident if the system has infinite number of solution
• parallel if the system has no solution.

Algebraic Methods of Solving Simultaneous Equations in Two Variables

Ways to solve simultaneous equations with two variables:-

1. Elimination Method

1. Substitution Method

2. Comparison Method

3. Elimination Method

2. Cramer’s Rule (Cross multiplication method)