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Expanding Single Brackets
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Foundational Skills
Draw the grid
\[ \text{Grid for } 3(x+5) \]
Set up the grid
Positive Multiplier
Integer × Two Terms
\[ 3(x+5) \rightarrow 3x+15 \]
Positive integer times two terms
Integer × (ax + b)
\[ 4(3x+5) \rightarrow 12x+20 \]
Coefficient on variable
Integer × like terms
\[ 3(2x+5x) \rightarrow 21x \]
Simplify result
Integer × diff vars
\[ 3(x+y) \rightarrow 3x+3y \]
Different variables
Integer × diff vars (coeff)
\[ 4(2x+3y) \rightarrow 8x+12y \]
Coefficients on variables
Negative Multiplier
Negative Integer
\[ -3(x+5) \rightarrow -3x-15 \]
Negative outside bracket
Neg Int × (ax + b)
\[ -4(3x+5) \rightarrow -12x-20 \]
Negative with coefficients
Negative Sign Only
\[ -(x+5) \rightarrow -x-5 \]
Implied multiplier of -1
Neg Int × diff vars
\[ -3(x+y) \rightarrow -3x-3y \]
Negative with diff vars
Neg Int × diff vars (coeff)
\[ -4(2x+3y) \rightarrow -8x-12y \]
Neg with diff vars coeffs
Variable Multiplier
Variable × Two Terms
\[ x(x+3) \rightarrow x^2+3x \]
Squared result
Variable × (ax + b)
\[ x(2x+5) \rightarrow 2x^2+5x \]
Squared with coeff
Coeff Var × Two Terms
\[ 2x(x+3) \rightarrow 2x^2+6x \]
Coeff outside bracket
Coeff Var × (ax + b)
\[ 2x(3x+5) \rightarrow 6x^2+10x \]
Coeff outside & inside
Neg Var × Two Terms
\[ -x(x+3) \rightarrow -x^2-3x \]
Negative variable
Neg Coeff Var
\[ -2x(x+3) \rightarrow -2x^2-6x \]
Negative coeff var
Three Term Brackets
Integer × 3 Terms
\[ 3(x+2+y) \]
Expanding 3 terms
Integer × 3 Vars
\[ 4(2x+3y+5z) \]
3 different variables
Integer × Like Terms
\[ 3(2x+5+4x) \]
Expand then simplify
Neg Int × 3 Terms
\[ -3(x+2+y) \]
Negative x 3 terms
Neg Sign × 3 Terms
\[ -(x+2+y) \]
Flip all signs
Special Cases
Var × (Same + Diff)
\[ x(x+y) \rightarrow x^2+xy \]
Mixed variables
Coeff Var × (Coeffs)
\[ 2x(3x+4y) \rightarrow 6x^2+8xy \]
Mixed with coeffs
Expand and Simplify
\[ 3(x+2)-6 \rightarrow 3x \]
Two-step problem
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