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Lesson · Grade 11 · Pre Calculus

Multiplying and Dividing Rational Expressions

  • Grade 11
  • Pre Calculus
  • 2 min 9 s
  • British Columbia
  • BC11-MA-113

The video factors (x² − 4)/(x + 3) × (x + 3)/(x − 2), writes the restrictions x ≠ −3 and x ≠ 2, and cancels the common factors to get x + 2. It then flips the divisor in (x² − 9)/(2x) ÷ (x + 3)/(4x²) to get 2x(x − 3) with x ≠ 0 and x ≠ −3, checks it at x = 1, and warns about the two common mistakes before a one-line recap.

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In one sentence

The question this lesson answers

A rational expression is a fraction of two polynomials. How do we multiply two of them?

First, factor everything. x squared minus 4 is a difference of squares, x squared minus 2 squared, so it is x minus 2 times x plus 2.

Transcript

What the lesson says

A rational expression is a fraction of two polynomials. How do we multiply two of them?

First, factor everything. x squared minus 4 is a difference of squares, x squared minus 2 squared, so it is x minus 2 times x plus 2.

A denominator can never be 0, so x cannot be negative 3 or 2. These are the restrictions.

Now cancel any factor that is in a numerator and a denominator, since each pair divides to 1: first x plus 3, then x minus 2.

What is left is x plus 2. The restrictions stay, because the original product is undefined at those values.

Now divide. The fraction we divide by, the second one, is called the divisor.

Restrictions first: the denominators 2x and 4x squared give x not equal to 0.

The divisor cannot be 0 either, so its numerator, x plus 3, gives x not equal to negative 3.

To divide, multiply by the reciprocal: flip the divisor upside down.

Factor x squared minus 9, a difference of squares: x minus 3 times x plus 3.

Cancel x plus 3. Then 4x squared over 2x simplifies to 2x.

So the answer is 2x times x minus 3, with x not equal to 0 or negative 3.

Check with x equals 1: negative 8 over 2, divided by 4 over 4, is negative 4, and so is 2 times 1 times negative 2.

Two common mistakes: flipping the wrong fraction, and missing the restriction from the divisor's numerator.

To recap: factor everything, find the restrictions, and cancel. To divide, flip the divisor and multiply.

Curriculum

Where this fits

British Columbia, Grade 11

Pre Calculus 11, Algebra: rational expressions and equationsBC11-MA-113

Previous: Factoring by Substitution

The outcomes and strands follow the British Columbia curriculum.

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