Multiplication and Division Revision Notes for Grade 12 NSC Matric Mathematics

Multiplication and Division

Understanding sign rules in multiplication

When working with algebraic expressions, it's essential to understand how signs behave during multiplication. These rules will help you determine whether your answer should be positive or negative.

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Sign rules for multiplication:

Positive × Positive = Positive
Positive × Negative = Negative
Negative × Positive = Negative
Negative × Negative = Positive

A simple way to remember this is: when the signs are the same, the result is positive. When the signs are different, the result is negative.

Working with division and fractions

Division by fractions follows a special rule that makes calculations much easier. When you divide by a fraction, you can change the operation to multiplication by using the reciprocal of the fraction.

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Key rule: Dividing by a fraction is the same as multiplying by its reciprocal.

The formula is: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

This means you "flip" the second fraction (swap the numerator and denominator) and change division to multiplication.

Multiplying algebraic fractions

When multiplying fractions that contain variables, follow these straightforward steps:

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Multiply the numerators together to get the new numerator
Multiply the denominators together to get the new denominator
Simplify the resulting fraction by cancelling common factors

Worked example 1: Basic fraction multiplication

Let's multiply: $\frac{6x}{7y} \times \frac{3}{5z}$

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Worked Example: Multiplying Algebraic Fractions

Step 1: Multiply numerators: $6x \times 3 = 18x$

Step 2: Multiply denominators: $7y \times 5z = 35yz$

Step 3: Write the result: $\frac{18x}{35yz}$

This fraction cannot be simplified further as there are no common factors.

Dividing algebraic fractions

Division of algebraic fractions becomes much simpler when you convert it to multiplication using the reciprocal method.

Worked example 2: Dividing algebraic fractions

Let's divide: $\frac{6x}{8y} \div \frac{3}{12z}$

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Worked Example: Dividing Algebraic Fractions

Step 1: Rewrite as multiplication by the reciprocal: $\frac{6x}{8y} \times \frac{12z}{3}$

Step 2: Multiply the fractions: $\frac{6x \times 12z}{8y \times 3} = \frac{72xz}{24y}$

Step 3: Simplify by finding common factors: $\frac{72xz}{24y} = \frac{3xz}{y}$ (dividing both numerator and denominator by 24)

The distributive law

The distributive law is a fundamental rule used to expand expressions where a single term multiplies a bracketed expression.

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Formula: $a(b + c) = ab + ac$

This means you multiply the term outside the bracket by each term inside the bracket separately.

Worked example 3: Using the distributive law

Expand: $-3x(5x - 6y)$

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Worked Example: Using the Distributive Law

Step 1: Multiply $-3x$ by the first term: $-3x \times 5x = -15x^2$

Step 2: Multiply $-3x$ by the second term: $-3x \times (-6y) = +18xy$

Step 3: Combine the results: $-15x^2 + 18xy$

Notice how the negative signs work: negative times negative gives positive.

Expanding binomials using FOIL

When multiplying two brackets (binomials), use the FOIL method. FOIL stands for First, Outer, Inner, Last - this tells you which terms to multiply together.

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Formula: $(a + b)(c + d) = ac + ad + bc + bd$

Worked example 4: FOIL method

Expand: $(2x + y)(3x - 2y)$

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Worked Example: FOIL Method

Step 1: First terms: $2x \times 3x = 6x^2$

Step 2: Outer terms: $2x \times (-2y) = -4xy$

Step 3: Inner terms: $y \times 3x = +3xy$

Step 4: Last terms: $y \times (-2y) = -2y^2$

Step 5: Combine all terms: $6x^2 - 4xy + 3xy - 2y^2$

Step 6: Simplify by combining like terms: $6x^2 - xy - 2y^2$

Essential exam tips

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Always check your signs carefully - this is where most errors occur
When dividing by fractions, remember to flip the second fraction and multiply
Use FOIL systematically to avoid missing terms when expanding brackets
Always look for opportunities to simplify your final answer
Combine like terms whenever possible after expanding

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Key Points to Remember:

Same signs multiply to give positive, different signs give negative
To divide by a fraction, multiply by its reciprocal instead
The distributive law lets you expand single terms over brackets: $a(b + c) = ab + ac$
Use FOIL (First, Outer, Inner, Last) to expand two brackets systematically
Always simplify your final answer by combining like terms and cancelling common factors

Multiplication and Division (Grade 12 NSC Matric Mathematics): Revision Notes