ALGEBRAIC FRACTIONS 6

Polynomial fractions also behave in a similar way to numerical fractions. Note their similarity in the following definitions and procedures.

Remember

A polynomial can be considered as an algebraic fraction with a denominator of 1. Observe

An algebraic fraction is irreducible if its numerator and denominator are relatively prime.

Mental arithmetic

1. Simplify these fractions:

We call the quotient of two polynomials an algebraic fraction:

For example, the following are algebraic fractions:

Simplification

If the numerator and the denominator of an algebraic fraction can be divided by the same polynomial (of a degree greater than or equal to 1), doing so will simplify the fraction.

If we divide the numerator and the denominator of an algebraic fraction by their GCD, we get an irreducible fraction.

Equivalent fractions

Two algebraic fractions are equivalent if:

• One of them is obtained by simplifying the other.

2. Say whether the fractions in each pair are equivalent or not:

• Both result in the same fraction when simplified.

For example, the following fractions are equivalent because both result in the same fraction when simplified:

Think and practise anayaeducacion.es Simplify algebraic fractions.

1 Simplify the following fractions:

2 Are the fractions in each pair equivalent? Check.

Mental arithmetic

1. Reduce the following to a common denominator.

Reduction to a common denominator

When we multiply the numerator and the denominator of an algebraic fraction by the same polynomial, we get an equivalent fraction.

Therefore, if we have several algebraic fractions, we can obtain others that, being respectively equivalent to the first ones, have the same denominator. When we do this, we say that we have reduced them to a common denominator.

x 1 3 1 2 and –2 +

2. Calculate. a) x x x 31 3 –2 +

Operations: addition, subtraction, multiplication and division

To add algebraic fractions, we reduce them to a common denominator and add the numerators together. Subtraction is a special type of addition.

The product of two algebraic fractions is equal to the product of their numerators divided by the product of their denominators.

b) x x 1 3 1 2 –2 + + c) · x x x x 2 24 –2 + d) : x x x x 25 5 –– 2 2 3 Perform the operations and simplify the result.

Inverse fractions. The inverse of x xx 31 5 ––2 is xx x 5 31 ––2 , as their product is a fraction equivalent to 1.

The quotient of two algebraic fractions is equal to the first multiplied by the inverse of the second.

anayaeducacion.es Operations with algebraic fractions. Breaking down algebraic fractions into simpler fractions

When multiplying and dividing algebraic fractions, sometimes it is easier to simplify them first. We do this by factorising the numerator and denominator. For example: x(x

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