Table of Contents
~~TOC~~
Lab46 Tutorials
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Shortcuts in Algebra
The aim of this page is to explore numbers and the patterns they create. The these examples demonstrate recognized methods of solving algebraic forms by manipulating the mathematic order of precedence.
The expressed goal of this exploration is to build heavily on the concepts of multiplying and dividing polynomials and to demonstrate basic mathematic algorithms that can be used to create workable solutions.
Distributive Law
Rule:
<latex>a(x-y)=ax-ay</latex>
<latex>a(x+y)=ax+ay</latex>
Derived
The distributive property can be explained using small numbers. Take for example, <latex>3 * 5 = 15</latex>. Both 3 and 5 can be written different ways. For example, 3 can be written as 2+1. Thus, our equation could be written <latex>5 * (2+1)</latex> or <latex>5(2 + 1)</latex>. In this case, the distributive property states that the value of a, in this case 5 should be distributed across both terms inside the parenthesis. Therefore, <latex>5(2 + 1)</latex> is equal to <latex>(5 * 2) + (5 * 1)</latex>
- 3 * 5 = 5(2+1) = (5 * 2) + (5 * 1) = 15
Example:
<latex>6(b-c)=6b-6c</latex>
<latex>5a(a-4c)=5a^2-20ac</latex>
FOIL
The Rule
- <latex>(a+b)(c+d) = ac + ad + bc + bd</latex>
Derived
FOIL is a mnemonic used to describe the process of multiplying two binomials. The process is as follows:
- First - The first two terms of the binomial are multiplied together
- Outside - The first term in the first binomial is multiplied by the second term in the second binomial
- Inside - The second term in the first binomial is multiplied by the first term in the second binomial
- Last - The second term in the first binomial is multiplied by the second term in the second binomial
This concept is more plainly derived from real numbers. Take this simple equation:
- <latex>6 * 8 = 48</latex>
The above equation states that 6 repeated 8 times is 48. However, 6 can also be written as 4+2, and the distributive law states that <latex>8(4+2)</latex> is equal to <latex>(8 * 4) + (8 * 2)</latex> which must also be equal to 48.
- <latex>6 * 8 = 8(4 + 2) = (8 * 4) + (8 * 2) = 48</latex>
Building on this concept, 8 can also be represented in different ways. 8 could be written as 7+1. So really, this example can be written in binomial notation:
- <latex>6 * 8 = (4 + 2)(7 + 1)</latex>
This is where the above rule comes into play. To multiply binomials, distribute each value across all other values. The above binomial, “foiled” out looks like this:
- <latex>6 * 8 = (4 + 2)(7 + 1) = (4 * 7) + (4 * 1) + (2 * 7) + (2 * 1) = 48</latex>
Difference of Squares
Rule:
<latex>(x+y)(x-y)=x^2-y^2</latex>
Derived
<latex>(x+y)(x-y) = x^2 + xy -xy - y^2 = x^2-y^2</latex>
Example:
<latex>(T+6)(T-6) = T^2 - 6T + 6T - 36 = T^2 - 36</latex>
<latex>(4x+5y)(4x-5y) = 16x^2 - 25y + 25y - 25y^2 = 16x^2 - 25y^2</latex>
Perfect Square Binomial
Rule:
<latex>(x+y)^2=x^2+2xy+y^2</latex>
<latex>(x-y)^2=x^2-2xy+y^2</latex>
Example:
<latex>(i+3)^2=i^2+6i+9</latex>
<latex>(3y-5)^2 \to 9y^2-2(3y)(5)+25 \to 9y^2-30y+25</latex>
Irregular Forms
Rule:
<latex>(x+a)(x+b)=x^2+x(a+b)+ab</latex>
Example:
Rule:
<latex>(ax+b)(cx+d)=acx^2+(ad+bc)+bd</latex>
Example:
<latex>(30x+7y)(2x-9y) \to 600x^2+(270y+140y)x-63y^2 \to 600x^2-130xy-63y^2</latex>
Prefect Cubes
<latex> (x-y)^3=x^3-3ax^2+3a^2x-a^3 </latex>
Sum of Cubes
<latex>x^3+y^3=(x+y)(x^2-xy+y^2)</latex>
Difference of Cubes
<latex>x^3-y^3=(x-y)(x^2+xy+y^2)</latex>
Quick Reference
Distributive Property
- <latex>a(x-y)=ax-ay</latex>
- <latex>a(x+y)=ax+ay</latex>
FOIL
- <latex>(a+b)(c+d) = ac + ad + bc + bd</latex>
Difference of Squares
- <latex>(x+y)(x-y)=x^2-y^2</latex>
Perfect Square Binomial
- <latex>(x+y)^2=x^2+2xy+y^2</latex>
- <latex>(x-y)^2=x^2-2xy+y^2</latex>
Perfect Cubes
- <latex>(x-y)^3=x^3-3ax^2+3a^2x-a^3</latex>
Sum of Cubes
- <latex>x^3+y^3=(x+y)(x^2-xy+y^2)</latex>
Difference of Cubes
- <latex>x^3-y^3=(x-y)(x^2+xy+y^2)</latex>
Irregular Forms
- <latex>(x+a)(x+b)=x^2+x(a+b)+ab</latex>
- <latex>(ax+b)(cx+d)=acx^2+(ad+bc)+bd</latex>