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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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]
Note: Remember two important points i.e.
First point involves a binomial [latex](a + b)^2[/latex]
A common mistake here is to conclude [latex](a + b)^2[/latex] = [latex]a^2 + b^2[/latex] when these two are not equal i.e.
[latex](a + b)^2[/latex] ≠ [latex]a^2 + b^2[/latex]
Similarly, second one is [latex](a - b)^2[/latex] ≠ [latex]a^2 - b^2[/latex]
To find [latex](a + b)^2[/latex], first write it as (a + b)(a +b)
i.e. [latex](a + b)^2[/latex] = (a + b)(a +b)
= [latex]a^2 + ab + ab + b^2[/latex]
= [latex]a^2 + 2ab + b^2[/latex]
The same technique is used to show [latex](a - b)^2[/latex] = [latex]a^2 - 2ab + b^2[/latex]
The above two products i.e. [latex](a + b)^2[/latex] and [latex](a - b)^2[/latex] are known as special products.
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.
- 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
- 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[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(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