Simplifying

Simplifying can refer to:

• Reducing a fraction.
• Reducing the number of terms.
• Reducing the number of factors.

Examples

Reducing the number of terms

Simplify:
$1*x + 3 - 4*x + 8$

Add terms.
=$5*x + 11$

Reducing the number of factors

Simplify:
$2*3*7*x$

Multiply factors.
=$42*x$

Multiple forms of simplification

Simplify:
$\frac{12*7*x-4*x}{3*2*x+5*x+9*x}$

Multiply factors.
=$\frac{84*x-4*x}{6*x+5*x+9*x}$

Add terms.
=$\frac{80*x}{20*x}$

Write the numerator and denominator as products of their prime factors:
=$\frac{2*2*2*2*5*x}{2*2*5*x}$

The common factors are 2, 2, 5 and x. Divide numerator and denominator by 2*2*5*x:
= $\frac{2*2*x}{1}$ = ${2*2*x}$

Multiply factors.
=$4 * x$

Expanding

The opposite of simplifying is expanding. Expansion can be:

• Seperating a number into powers of ten, or more simply, seperating numbers into their digits. Expanded form of numbers helps conceptualize the significance of each place value.
• Distributing a factored polynomial into a sum of unlike terms. The latter definition is used more frequently.

Determining Expanded Expressions

An algebraic expression is expanded if all possible like terms are combined and the expression follows the order of operations without brackets (multiplication will always come before subtraction). The following is an expanded expression.

$1 + a + 3b - 5c^2 + \frac d3 - \frac 1{e^f} + \sqrt {ln(g)}$

Expanding Algebraic Expressions

$\biggl(\frac {a+b}c\biggr)^2 = \frac {(a+b)^2}{c^2} = \frac {a^2+2ab+b^2}{c^2} = \frac{a^2}{c^2} + \frac{2ab}{c^2} + \frac{b^2}{c^2}$

Try to expand this:

$\frac {-2(x-9)}6 + \frac {9(2b^2-x-y)}7 + \frac {b(b-3)}2$

Distributing factors:

$\frac {-2x+18}6 + \frac {18b^2-9x-9y}7 + \frac {b^2-3b}2$

Adding fractions (you could also distribute the denominators):

$\frac {-14x + 126 + 108b^2 - 54x - 54y + 21b^2 - 63b}{42}$

Combining like terms:

$\frac {-68x+126+129b^2-54y-63b}{42}$

Distributing the common denominator:

$\frac {-34}{21}x + 3 + \frac {43}{14}b^2 - \frac 97y - \frac 32b$