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Expanding Double and Triple Brackets
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Foundational skills
Identify terms in an expression
\[ x + 5 \text{ has 2 terms} \]
Count the number of terms.
Set up a 2×2 grid
\[ (x + 4)(x + 2) \]
Identify row and column labels.
Fill in one cell of a grid
\[ x \times 5 = 5x \]
Multiply two terms for a grid cell.
Collect like terms
\[ x^2 + 3x + 5x + 15 \]
Combine like terms to simplify.
Double brackets: Both terms positive
Expand (x + a)(x + b) basic
\[ (x + 3)(x + 4) \]
Both constants positive.
Expand with larger constants
\[ (x + 7)(x + 8) \]
Larger single digit constants.
Double brackets: Both terms negative
Expand (x − a)(x − b) basic
\[ (x – 2)(x – 5) \]
Both constants subtracted.
Expand with larger negatives
\[ (x – 6)(x – 7) \]
Larger negative constants.
Double brackets: Mixed signs
Expand (x + a)(x − b)
\[ (x + 4)(x – 3) \]
Positive first, negative second.
Expand (x − a)(x + b)
\[ (x – 3)(x + 7) \]
Negative first, positive second.
Difference of two squares
\[ (x + 5)(x – 5) \]
Middle terms cancel out.
Expand (x − a)(x − b) both negative
\[ (x – 3)(x – 5) \]
Both constants are negative.
Double brackets: Squared brackets
Expand (x + a)²
\[ (x + 4)^2 \]
Bracket squared, positive constant.
Expand (x − a)²
\[ (x – 3)^2 \]
Bracket squared, negative constant.
Double brackets: Coefficients on x
Expand (ax + b)(x + c)
\[ (2x + 3)(x + 4) \]
First bracket has coefficient.
Expand (x + a)(bx + c)
\[ (x + 5)(3x + 2) \]
Second bracket has coefficient.
Expand (ax + b)(cx + d)
\[ (2x + 3)(3x + 5) \]
Both brackets have coefficients.
Expand (ax − b)(cx + d)
\[ (3x – 2)(2x + 5) \]
Coefficients with mixed signs.
Expand (ax + b)²
\[ (2x + 3)^2 \]
Squared with coefficient on x.
Expand (ax − b)²
\[ (3x – 4)^2 \]
Squared with coefficient, subtraction.
Double brackets: Difference of two squares
Expand (ax − b)(ax + b)
\[ (2x – 3)(2x + 3) \]
Difference of squares with coefficient.
Double brackets: Constant first
Expand (a + bx)(c + dx)
\[ (3 + 2x)(5 + x) \]
Constant before variable.
Expand (a − bx)(c + dx)
\[ (4 – x)(2 + 3x) \]
Constant first, mixed signs.
Triple brackets: Basic form
Expand (x+a)(x+b)(x+c)
\[ (x+1)(x+2)(x+3) \]
Three brackets, all positive.
Expand (x+a)(x+b)(x−c)
\[ (x+2)(x+4)(x-1) \]
One negative constant.
Expand (x+a)(x−b)(x−c)
\[ (x+3)(x-2)(x-4) \]
Two negative constants.
Expand (x−a)(x−b)(x−c)
\[ (x-1)(x-2)(x-3) \]
All negative constants.
Triple brackets: With coefficients
Expand (x+a)(x+b)(cx+d)
\[ (x+1)(x+3)(2x+1) \]
Third bracket has coefficient.
Expand (ax+b)(x+c)(x+d)
\[ (2x+1)(x+2)(x+3) \]
First bracket has coefficient.
Triple brackets: Squared times linear
Expand (x + a)²(x + b)
\[ (x + 2)^2(x + 3) \]
Squared bracket times linear.
Expand (x − a)²(x + b)
\[ (x – 3)^2(x + 2) \]
Squared with subtraction.
Triple brackets: Cubed brackets
Expand (x + a)³
\[ (x + 2)^3 \]
Bracket cubed, positive constant.
Expand (x − a)³
\[ (x – 2)^3 \]
Bracket cubed, negative constant.
Special cases
Two brackets added
\[ (x+2)(x+3) + (x+1)(x+4) \]
Expand then add results.
Two brackets subtracted
\[ (x+5)(x+2) – (x+1)(x+3) \]
Expand then subtract results.
Show (x+a)² ≠ x²+a²
\[ (x + 3)^2 \neq x^2 + 9 \]
Prove the middle term exists.
Double minus single bracket
\[ (x+3)(x+4) – 2(x+5) \]
Expand both, then combine.
Coefficient outside brackets
\[ 2(x + 3)(x + 4) \]
Expand then multiply by constant.
Expand then subtract constant
\[ (x+a)(x+b) – ab \]
Expand then subtract a constant.
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