Quantitative Aptitude - SPLessons

Expanding Expressions

Chapter 54

SPLessons 5 Steps, 3 Clicks
5 Steps - 3 Clicks

Expanding Expressions

shape Introduction

Concepts covered in this chapter are like how to multiply expressions, expanding, multiplying monomials, multiplying the monomial by polynomials, foil method.


shape Methods

Multiplying monomials

Multiplying monomial by a monomial

Rule: Multiply members of the same “family”.


Examples:

(3x)(7x) = \(21x^2\)

\((5y^3)(4y^4)\) = \(20y^7\)

\((8k^5)(k) \) = \(8k^6\)

\( (6x^4y^3)(x^6y^2) \) = \(6x^(10)y^5\)

\( (a^2bc^3)(3b^3)(4a^5b) \) = \(12a^7b^5c^3\)


Multiplying monomials by polynomials

Rule: Multiply each term in the parentheses by the term in front.


Examples:

3(2x + 5) = 6x + 15

\(7x(4x^2 – 6x + 1) \) = \(28x^3 – 42×62 + 7x\)

\(-2x^2y(3x^3y – 9y^5) \) = \(-6x^5y^2 + 18x62y^6\)


This process of multiplying the term inside parentheses by the term in front is called expanding.

Simplify the following:

1. 3(2x + 1) + 5(x + 4)

= 6x + 3 + 5x + 20

= 11x + 23


2. 7(2 – y) – 2y(y – 3)

= 14 – 7y – 2\(y^2\) + 6y

= 14 – y – 2\(y^2\)

= -2\(y^2\) – y + 14


3. 3ab(2b – a) – (3\(a^2\)b + 6\(ab^2\))

= 3ab(2b – a) – 1(3\(a^2\)b + 6\(ab^2\))

= \(6ab^2 – 3a^2b – 3a^2b – 6ab^2\)

= \(-6a^2b\)


Multiplying two binomials

For multiplying two binomials, one of the method used that is known as foil method.

FOIL stands for

First

Outer

Inner

Last

Goal is to multiply each of the two terms in one binomial by each of the two terms in another binomial, whenever two binomials are multiplied.

To perform multiplications, Foil method is a systematic way.


Examples:

1. (x + 2)(x + 7)

= \(x^2\) + 7x + 2x + 14

= \(x^2\) +9x + 14


2. (3y – 4)(2y – 5)

= 6\(y^2\) – 15y – 8y + 20

= 6\(y^2\) -23y + 20


3. (2x + y)(x – 7y)

= 2\(x^2\) – 14xy + xy – 7\(y^2\)

= 2\(x^2\) – 13xy – 7\(y^2\)


shape Samples

1. Simplify the following
5(3x – 4) + 3(x – 5) + 4\(x^2\)

Solution:

    5(3x – 4) + 3(x – 5) + 4\(x^2\)

    15x – 20 + 3x – 15 + 4\(x^2\)

    = 4\(x^2\) + 18x – 35

    Therefore, 5(3x – 4) + 3(x – 5) + 4\(x^2\) = 4\(x^2\) + 18x – 35.


2. Simplify the following
10(\(x^2\) + 5) + 9(3\(x^2\) + 4x + 5) + 10

Solution:

    10(\(x^2\) + 5) + 9(3\(x^2\) + 4x + 5) + 10

    10\(x^2\) + 50 + 27\(x^2\) + 36x + 45 + 10

    = 37\(x^2\) + 36x + 105

    Therefore, 10(\(x^2\) + 5) + 9(3\(x^2\) + 4x + 5) + 10 = 37\(x^2\) + 36x + 105


3. Simplify the following
-3x(-3x – 4) + 4x(7x – 9) + 4\(x^2\)

Solution:

    3x(-3x – 4) + 4x(7x – 9) + 4\(x^2\)

    9\(x^2\) + 12x + 28\(x^2\) – 36x + 4\(x^2\)

    = 41\(x^2\) – 24x

    Therefore, -3x(-3x – 4) + 4x(7x – 9) + 4\(x^2\) = 41\(x^2\) – 24x


4. Simplify the following
4\(x^2\)(5x + 6) + 2x(\(x^2\) + 10x – 12)

Solution:

    4\(x^2\)(5x + 6) + 2x(\(x^2\) + 10x – 12)

    = 20\(x^3\) + 24\(x^2\) + 2\(x^3\) + 20\(x^2\) – 24x

    = 22\(x^3\) + 44\(x^2\) – 24x

    Therefore, 4\(x^2\)(5x + 6) + 2x(\(x^2\) + 10x – 12) = 22\(x^3\) + 44\(x^2\) – 24x


5. Simplify the following
5(6\(x^3\) + 5x + 7) + 3\(x^2\)

Solution:

    5(6\(x^3\) + 5x + 7) + 3\(x^2\)

    = 30\(x^3\) + 25x + 35 + 3\(x^2\)

    = 30\(x^3\) + 3\(x^2\) + 25x + 35

    Therefore, 5(6\(x^3\) + 5x + 7) + 3\(x^2\) = 30\(x^3\) + 3\(x^2\) + 25x + 35