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Simplifying, multiplying and dividing

Rational expressions

Japanese school year: Math II

What you learn

Learn to factor the numerator and denominator of a rational expression, cancel common factors, and multiply and divide such expressions. It is the basis for tidying expressions in equations and calculus. It helps to know factoring and how to multiply and divide fractions first.

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Key points

A factor common to the numerator and the denominator can be cancelled. Only factors (things multiplied) can be cancelled, never terms that are added.

ACBC=AB\frac{AC}{BC} = \frac{A}{B}

Before cancelling, factor the numerator and the denominator. Differences of squares and quadratics are the usual cases.

x2−a2=(x+a)(x−a)x^2 - a^2 = (x + a)(x - a)

To multiply, multiply the numerators and the denominators. To divide, flip the fraction you divide by and multiply.

AB×CD=ACBD,AB÷CD=AB×DC\frac{A}{B} \times \frac{C}{D} = \frac{AC}{BD}, \quad \frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \times \frac{D}{C}
Worked example

Choose the expression equal to this one.

x2+5x+6x2+3x+2\frac{x^{2} + 5x + 6}{x^{2} + 3x + 2}
x2+5x+6x2+3x+2{{}\frac{x^{2} + 5x + 6}{x^{2} + 3x + 2}}=(x+2)(x+3)(x+1)(x+2){{}= \frac{\left(x + 2\right)\left(x + 3\right)}{\left(x + 1\right)\left(x + 2\right)}}
=x+3x+1{{}= \frac{x + 3}{x + 1}}
Answer: x+3x+1\frac{x + 3}{x + 1}

Choose a set to practice.