Quantitative Aptitude - SPLessons

Expanding Expressions

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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.

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shape Methods

Multiplying monomials
Multiplying monomial by a monomial
Rule: Multiply members of the same “family”.
Examples:
(3x)(7x) = [latex]21x^2[/latex]
[latex](5y^3)(4y^4)[/latex] = [latex]20y^7[/latex]
[latex](8k^5)(k) [/latex] = [latex]8k^6[/latex]
[latex] (6x^4y^3)(x^6y^2) [/latex] = [latex]6x^(10)y^5[/latex]
[latex] (a^2bc^3)(3b^3)(4a^5b) [/latex] = [latex]12a^7b^5c^3[/latex]
Multiplying monomials by polynomials
Rule: Multiply each term in the parentheses by the term in front.
Examples:
3(2x + 5) = 6x + 15
[latex]7x(4x^2 – 6x + 1) [/latex] = [latex]28x^3 – 42x62 + 7x[/latex]
[latex]-2x^2y(3x^3y – 9y^5) [/latex] = [latex]-6x^5y^2 + 18x62y^6[/latex]
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[latex]y^2[/latex] + 6y
= 14 – y – 2[latex]y^2[/latex]
= -2[latex]y^2[/latex] – y + 14
3. 3ab(2b - a) – (3[latex]a^2[/latex]b + 6[latex]ab^2[/latex])
= 3ab(2b - a) – 1(3[latex]a^2[/latex]b + 6[latex]ab^2[/latex])
= [latex]6ab^2 – 3a^2b – 3a^2b - 6ab^2[/latex]
= [latex]-6a^2b[/latex]
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)
= [latex]x^2[/latex] + 7x + 2x + 14
= [latex]x^2[/latex] +9x + 14
2. (3y - 4)(2y - 5)
= 6[latex]y^2[/latex] – 15y – 8y + 20
= 6[latex]y^2[/latex] -23y + 20
3. (2x + y)(x – 7y)
= 2[latex]x^2[/latex] – 14xy + xy – 7[latex]y^2[/latex]
= 2[latex]x^2[/latex] – 13xy - 7[latex]y^2[/latex]

shape Samples

1. Simplify the following 5(3x - 4) + 3(x - 5) + 4[latex]x^2[/latex]
Solution:
    5(3x - 4) + 3(x - 5) + 4[latex]x^2[/latex]
    15x - 20 + 3x - 15 + 4[latex]x^2[/latex]
    = 4[latex]x^2[/latex] + 18x – 35
    Therefore, 5(3x - 4) + 3(x - 5) + 4[latex]x^2[/latex] = 4[latex]x^2[/latex] + 18x – 35.

2. Simplify the following 10([latex]x^2[/latex] + 5) + 9(3[latex]x^2[/latex] + 4x + 5) + 10
Solution:
    10([latex]x^2[/latex] + 5) + 9(3[latex]x^2[/latex] + 4x + 5) + 10
    10[latex]x^2[/latex] + 50 + 27[latex]x^2[/latex] + 36x + 45 + 10
    = 37[latex]x^2[/latex] + 36x + 105
    Therefore, 10([latex]x^2[/latex] + 5) + 9(3[latex]x^2[/latex] + 4x + 5) + 10 = 37[latex]x^2[/latex] + 36x + 105

3. Simplify the following -3x(-3x - 4) + 4x(7x - 9) + 4[latex]x^2[/latex]
Solution:
    3x(-3x - 4) + 4x(7x - 9) + 4[latex]x^2[/latex]
    9[latex]x^2[/latex] + 12x + 28[latex]x^2[/latex] - 36x + 4[latex]x^2[/latex]
    = 41[latex]x^2[/latex] - 24x
    Therefore, -3x(-3x - 4) + 4x(7x - 9) + 4[latex]x^2[/latex] = 41[latex]x^2[/latex] - 24x

4. Simplify the following 4[latex]x^2[/latex](5x + 6) + 2x([latex]x^2[/latex] + 10x - 12)
Solution:
    4[latex]x^2[/latex](5x + 6) + 2x([latex]x^2[/latex] + 10x - 12)
    = 20[latex]x^3[/latex] + 24[latex]x^2[/latex] + 2[latex]x^3[/latex] + 20[latex]x^2[/latex] - 24x
    = 22[latex]x^3[/latex] + 44[latex]x^2[/latex] - 24x
    Therefore, 4[latex]x^2[/latex](5x + 6) + 2x([latex]x^2[/latex] + 10x - 12) = 22[latex]x^3[/latex] + 44[latex]x^2[/latex] - 24x

5. Simplify the following 5(6[latex]x^3[/latex] + 5x + 7) + 3[latex]x^2[/latex]
Solution:
    5(6[latex]x^3[/latex] + 5x + 7) + 3[latex]x^2[/latex]
    = 30[latex]x^3[/latex] + 25x + 35 + 3[latex]x^2[/latex]
    = 30[latex]x^3[/latex] + 3[latex]x^2[/latex] + 25x + 35
    Therefore, 5(6[latex]x^3[/latex] + 5x + 7) + 3[latex]x^2[/latex] = 30[latex]x^3[/latex] + 3[latex]x^2[/latex] + 25x + 35
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