Simplify, multiply and divide algebraic fractions

For x nonzero, which is the fully simplified expression?

Original

\(\displaystyle \frac{(x+2)(x+3)}{(x+2)(x+5)}\)

Whole factor cancelled

\(\displaystyle \frac{x+3}{x+5}\)

Terms cancelled

\(\displaystyle \frac{3}{5}\)

Matching input

\(\displaystyle x=0\)

Other inputs to try

\(\displaystyle x=2,\quad x=8\)

Compare the original expression, the result of cancelling a whole factor, and the result of wrongly cancelling terms. They agree at zero. Try two and eight. Values are rounded to three decimal places

Which comparison holds throughout the slider range?

Input

\(\displaystyle x=0\)

Original

\(\displaystyle \frac{(x+2)(x+3)}{(x+2)(x+5)}\)

Whole factor cancelled

\(\displaystyle \frac{x+3}{x+5}\)

Terms cancelled

\(\displaystyle \frac{3}{5}\)

The three values matched at x = 0. What does that single match establish?

Cancellation preserves restrictions

A denominator cannot be zero. Removing a factor from the written expression does not make an originally excluded value valid

1. 1Cancel factors Factorise before cancelling. Divide both numerator and denominator by the same nonzero factor \(\displaystyle \frac{6x(x+4)}{9x}=\frac{2(x+4)}{3}\)
2. 2Retain the original domain Here the original denominator is 9x, so x remains nonzero \(\displaystyle x\ne0\)
3. 3Do not cancel terms The x in x + 4 cannot be removed on its own

Which value of x is excluded because x + 2 is a denominator factor?

Multiply the fractions and simplify

For x nonzero, what exponent of x remains after simplifying this product?

For x nonzero, choose the complete simplified result

Division uses a reciprocal

Keep the first fraction unchanged. Reverse only the divisor, then multiply and cancel

1. 1Change the operation The denominators b and d must be nonzero. The divisor must also be nonzero, so c is nonzero \(\displaystyle \frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}\)
2. 2Record restrictions first Keep every exclusion from the original fractions, including factors that disappear after cancellation

For x nonzero, what should multiply the first fraction when this division is rewritten?

For x nonzero, evaluate the simplified quotient

Quotient

\(\displaystyle \frac{6x}{5}\div\frac{3x}{10}\)

Compare the quotient below with the incorrect product that leaves the divisor unchanged. Move x through positive values. Values are rounded to two decimal places

Why does the quotient stay at 4?

Which values must be excluded from the original quotient?

For x not equal to 2 and y nonzero, simplify this product

Opposite factors differ by a sign

Reversing the order of a subtraction changes the sign. Rewrite one factor before cancelling

1. 1Opposite order Multiplying by negative one reverses both signs \(\displaystyle 4-x=-(x-4)\)
2. 2The ratio is negative one The denominator still excludes x = 4. \(\displaystyle \frac{4-x}{x-4}=-1\)

For x not equal to 4, simplify the fraction

Substitute x = -1 into the original fraction. What value do you obtain?

At x = 0 the simplified formula gives the value below. What can you conclude about the original fraction?