Algebra for the Practical Man

Simplify Equations

When simplifying an equation organize them in alphabetical order with the exponent in descending order. If any conflict is set use the exponent as a rule of thumb to order the equation.

Multiplying

Monomials When multiplying monomials.

If multiplying by symbol or literal numbers, the coefficients never change and only the letters are multiplied. $a \times (-3b^4) = -3ab^4$

If both terms has numerical coefficients, multiply both than multiply the literal part like the previous one: $(4x) . (7y^2) = 28xy^2$

When multiplying numbers with exponent, just sum the original exponents: $(4x^2) . (7x^2) = 28x^4$

Polynomial

$4(5-2) = 4 \times 3 = 12$

$(4.2) - (4.5) = 4 \times 3 = 12$

The product of any two polynomials is found by multiplying each term of one by each term of the other and adding the result:

$(2a + c)(x - 3y + 5d) = 2ax - 6ay + 10ad + cx - 3cy + 5cd$

The square of the SUM of two terms is the sum of their separate squares PLUS twice their product, and the square of the difference of two terms is the sum of their square MINUS twice their product.

$(x + y)^2 = x^2 + y^2 + 2xy$

$(x - y)^2 = x^2 + y^2 - 2xy$

The product of the sum and difference of two terms is equals the difference of their squares.

$(x+y)(x-y) = x^2 - y^2$

Dividing

Monomials To find the quotient of two monomials, divide coefficients, cancel commmon literal factors. If the same letter with different exponent appear, take the difference of the exponent.

$\frac{18x^2y^6z}{6x^6y^2} = \frac{3y^4}{x^4}$

Polynomials To divide polynomials is used an algorithm called Long Division.

Step 1: Divide the first term of the dividend (6x3) by the first term of the divisor (x2), and put that as the first term in the quotient (6x).
Step 2: Multiply the divisor by that answer, place the product (6x3 + 24x2 + 18x) below the dividend.
Step 3: Subtract to create a new polynomial (-12x2 - 16x + 25).
Step 4: Repeat the same process with the new polynomial obtained after subtraction.

Exponents laws

$a^m \times a^n = a^{m+n}$
$\frac{a^m}{a^n} = a^{m-n}$
$(a^n)^m = a^{m.n}$
$(ab)^m = a^m b^m$
$(\frac{a}{b})^m = \frac{a^m}{b^m}$
$a^0 = 1$
$\frac{1}{a^{-n}} = a^n$
$a^{1/m} = \sqrt[m]{a}$
$(\sqrt[n]{a})^m = \sqrt[n]{a^m}$
$a^{m/n} = \sqrt[n]{a^m}$
$\sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}$
$\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$

Equations

An equation is said to be a conditional equation or equation of condition when it fulfills or satisfies the condition expressed, eg:

$4x + 1 = 8x - 7$ when $x = 2$

The two parts of the equation that are connected by the equal sign are called members of the equation.

Rules

Any term of an equation can be transposed to the opposite member changing their sign.
Transpose the equation so that all terms which contain the unknow quantity are in the first member and all other terms are in the second member.
The begin of the alphabet is used to describe know variables, as ($a, b, c$) and the ending of the alphabet to describe unknow variables, as ($x, y$).

An equation of any degree is said to be solved when a value of the unknow is found which satisfies the equation. Quadratic equations and beyond can have more than one solution.

Exponential equations is when you have an unknow value as exponent.

Good links:

https://algebrarules.com