How to Multiply Rational Fractions With Two Variables

Students learn to multiply fractions with variables during a first-year algebra course.
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A rational fraction is any fraction in which the denominator doesn’t equal zero. In algebra, rational fractions possess variables, which are unknown quantities represented by letters of the alphabet. Rational fractions can be monomials, possessing one term each in the numerator and denominator, or polynomials, with multiple terms in the numerator and denominator. As with arithmetic fractions, most students find multiplying algebraic fractions a simpler process than adding or subtracting them.

Monomials

    Multiply the coefficients and constants in the numerator and denominator separately. Coefficients are numbers attached to the left-hand sides of the variables, and constants are numbers without variables. For instance, consider the problem (4x2)/(5y) * (3)/(8xy3). In the numerator, multiply 4 by 3 to get 12, and in the denominator, multiply 5 by 8 to get 40.

    Multiply the variables and their exponents in the numerator and denominator separately. When multiplying powers that have the same base, add their exponents. In the example, no multiplication of variables occurs in the numerators, because the second fraction’s numerator lacks variables. So, the numerator remains x2. In the denominator, multiply y by y3, obtaining y4. Hence, the denominator becomes xy4.

    Combine the results of the previous two steps. The example produces (12x2)/(40xy4).

    Reduce the coefficients to lowest terms by factoring out and canceling the greatest common factor, just as you would in a non-algebraic fraction. The example becomes (3x2)/(10xy4).

    Reduce the variables and exponents to lowest terms. Subtract smaller exponents on one side of the fraction from the exponents of their like variable on the opposite side of the fraction. Write the remaining variables and exponents on the side of the fraction that initially possessed the larger exponent. In (3x2)/(10xy4), subtract 2 and 1, the exponents of x terms, getting 1. This renders x^1, normally written just x. Place it in the numerator, since it originally possessed the greater exponent. So, the answer to the example is (3x)/(10y4).

Polynomials

    Factor the numerators and denominators of both fractions. For example, consider the problem (x2 + x – 2)/(x2 + 2x) * (y – 3)/(x2 – 2x + 1). Factoring produces [(x – 1)(x + 2)]/[x(x + 2)] * (y – 3)/[(x – 1)(x – 1)].

    Cancel and cross-cancel any factors shared by both the numerator and denominator. Cancel terms top-to-bottom in individual fractions as well as diagonal terms in opposite fractions. In the example, the (x + 2) terms in the first fraction cancel, and the (x – 1) term in the numerator of the first fraction cancels one of the (x – 1) terms in the denominator of the second fraction. Thus, the only remaining factor in the numerator of the first fraction is 1, and the example becomes 1/x * (y – 3)/(x – 1).

    Multiply the numerator of the first fraction by the numerator of the second fraction, and multiply the denominator of the first by the denominator of the second. The example yields (y – 3)/[x(x – 1)].

    Expand any terms left in factored form, eliminating all parentheses. The answer to the example is (y – 3)/(x2 – x), with the constraint that x cannot equal 0 or 1.

    Tips

    • In order to multiply polynomial fractions, you must first know how to factor and expand. When multiplying monomial fractions you can also cross-cancel, which essentially amounts to simplifying before multiplication by reducing the diagonals of the fraction.

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