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Multiplying and Dividing Algebraic Terms
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Foundational Skills
Number next to variable means multiply
\[ 5y \text{ means } 5 \times y \]
Recognise notation for multiplication.
Power notation for variables
\[ x^3 = x \times x \times x \]
Understand what powers mean.
Identify the coefficient
\[ \text{In } 7x \text{, coef} = 7 \]
Find the number part of a term.
Coefficient with a power
\[ \text{In } 4x^2 \text{, coef} = 4 \]
Find coefficient when there’s a power.
Multiplying by Integers
Integer × variable
\[ 5 \times x = 5x \]
Multiply a number by a variable.
Integer × term with coefficient
\[ 3 \times 4y = 12y \]
Multiply a number by an algebraic term.
Term × integer
\[ 5x \times 2 = 10x \]
Multiply an algebraic term by a number.
Multiplying Variables
Variable × itself
\[ x \times x = x^2 \]
Multiply a variable by itself.
Different variables
\[ x \times y = xy \]
Multiply two different variables.
Variable × power
\[ x \times x^2 = x^3 \]
Multiply a variable by a power term.
Power × power
\[ x^2 \times x^3 = x^5 \]
Multiply two power terms.
Multiplying Terms
Terms → squared answer
\[ 3x \times 4x = 12x^2 \]
Multiply terms to get a squared result.
Terms with different variables
\[ 3x \times 4y = 12xy \]
Multiply terms with different variables.
One term has a power
\[ 2x^2 \times 3x = 6x^3 \]
Multiply where one term has a power.
Both terms have powers
\[ 2x^2 \times 3x^2 = 6x^4 \]
Multiply terms that both have powers.
Two different variables
\[ 2xy \times 3x = 6x^2y \]
Multiply terms with two variables.
Square a term
\[ (3x)^2 = 9x^2 \]
Square both coefficient and variable.
Cube a term
\[ (2x)^3 = 8x^3 \]
Cube both coefficient and variable.
Dividing by Integers
Term ÷ integer
\[ \frac{12x}{4} = 3x \]
Divide an algebraic term by a number.
Power term ÷ integer
\[ \frac{8x^2}{2} = 4x^2 \]
Divide a term with a power by a number.
Two-variable term ÷ integer
\[ \frac{12xy}{3} = 4xy \]
Divide a two-variable term by a number.
Dividing by Variables
Term ÷ variable → number
\[ \frac{6x}{x} = 6 \]
Divide a term by its variable.
Power ÷ variable
\[ \frac{x^3}{x} = x^2 \]
Divide a power by the variable.
Power ÷ power
\[ \frac{x^5}{x^2} = x^3 \]
Divide a higher power by a lower power.
Two-var term ÷ one variable
\[ \frac{6xy}{y} = 6x \]
Divide by one of two variables.
Power term ÷ lower power
\[ \frac{6x^3}{x^2} = 6x \]
Divide by a lower power of the variable.
Dividing Term by Term
Terms ÷ → number
\[ \frac{12x}{3x} = 4 \]
Divide two terms to get a number.
Different powers
\[ \frac{12x^3}{4x} = 3x^2 \]
Divide terms with different powers.
Two variables
\[ \frac{12xy}{4x} = 3y \]
Divide where one variable cancels.
Same variable, higher powers
\[ \frac{15x^4}{3x^2} = 5x^2 \]
Divide terms both with powers.
Multiple variables and powers
\[ \frac{18x^2y}{6xy} = 3x \]
Divide with multiple variables.
Dividing Expressions
Two-term expression ÷ integer
\[ \frac{6x + 9}{3} = 2x + 3 \]
Divide each term by the same number.
Different variables
\[ \frac{10x + 15y}{5} \]
Divide expression with two variables.
With a power term
\[ \frac{6x^2 + 9x}{3} \]
Divide expression with a power term.
Three-term expression
\[ \frac{12x + 8y + 4}{4} \]
Divide each of three terms.
Powers with subtraction
\[ \frac{8x^2 – 12x}{4} \]
Divide expression with subtraction.
Special Cases
Answer has coefficient 1
\[ \frac{5x}{5} = x \]
Recognise when result is just the variable.
Terms ÷ → just a number
\[ \frac{8x}{2x} = 4 \]
Variable cancels completely.
Identify equivalent products
\[ 3x \times 4x = \text{?} \]
Choose correct simplified form.
Identify equivalent quotients
\[ \frac{12x^2}{4x} = \text{?} \]
Choose correct simplified form.
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