Algebraic Expansion, Factorisation, and Rearranging Formulas

Expansion of Algebraic Expressions

  • Foundational Principles of Expansion:

    • Expansion is the process of removing brackets (parentheses) by multiplying each term inside by the terms outside or by each term of another bracket, following the distributive property: a(b+c)=ab+aca(b + c) = ab + ac.
    • For binomial multiplication (a+b)(c+d)(a + b)(c + d), the FOIL mnemonic represents the systematic sum of products:
      • First: Multiply the first terms in each set of parentheses (a×ca \times c).
      • Outside: Multiply the outermost terms (a×da \times d).
      • Inside: Multiply the innermost terms (b×cb \times c).
      • Last: Multiply the last terms in each set of parentheses (b×db \times d).
      • Sum of terms: (a+b)(c+d)=ac+ad+bc+bd(a + b)(c + d) = ac + ad + bc + bd.

  • Expansion of Two Linear Binomials:

    • Problem 1: Expand and simplify (x+2)(x+1)(x + 2)(x + 1)
      • Application of FOIL:
        • First terms: x×x=x2x \times x = x^2
        • Outside terms: x×1=xx \times 1 = x
        • Inside terms: 2×x=2x2 \times x = 2x
        • Last terms: 2×1=22 \times 1 = 2
      • Unsimplified sum: x2+x+2x+2x^2 + x + 2x + 2
      • Combine like terms (x+2x=3xx + 2x = 3x):
        • Final simplified expression: x2+3x+2x^2 + 3x + 2
      • Common error note: Erroneously recording the result as x2+2x+3x^2 + 2x + 3 arises from failing to combine like terms correctly and transposing the constant and coefficient.
    • Problem 2: Expand and simplify (x−1)(x−11)(x - 1)(x - 11)
      • Application of FOIL:
        • First terms: x×x=x2x \times x = x^2
        • Outside terms: x×(−11)=−11xx \times (-11) = -11x
        • Inside terms: −1×x=−x-1 \times x = -x
        • Last terms: (−1)×(−11)=+11(-1) \times (-11) = +11
      • Unsimplified sum: x2−11x−x+11x^2 - 11x - x + 11
      • Combine like terms (−11x−x=−12x-11x - x = -12x):
        • Final simplified expression: x2−12x+11x^2 - 12x + 11
      • Common error note: Erroneously recording x2+10x+11x^2 + 10x + 11 results from sign errors when combining −11x-11x and −x-x.
    • Problem 5: Expand and simplify (4x+5)(4x+3)(4x + 5)(4x + 3)
      • Application of FOIL:
        • First terms: 4x×4x=16x24x \times 4x = 16x^2
        • Outside terms: 4x×3=12x4x \times 3 = 12x
        • Inside terms: 5×4x=20x5 \times 4x = 20x
        • Last terms: 5×3=155 \times 3 = 15
      • Unsimplified sum: 16x2+12x+20x+1516x^2 + 12x + 20x + 15
      • Combine like terms (12x+20x=32x12x + 20x = 32x):
        • Final simplified expression: 16x2+32x+1516x^2 + 32x + 15

  • Squaring Binomials (Perfect Squares):

    • Identities for squaring a binomial:
      • (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2
      • (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2
    • Problem 3: Expand and simplify (7+x)2(7 + x)^2
      • Rewrite as repeated multiplication: (7+x)(7+x)(7 + x)(7 + x)
      • Apply the identity with a=7a = 7 and b=xb = x:
        • a2=72=49a^2 = 7^2 = 49
        • 2ab=2×7×x=14x2ab = 2 \times 7 \times x = 14x
        • b2=x2b^2 = x^2
      • Unsimplified sum: 49+14x+x249 + 14x + x^2
      • Writing in descending powers of xx:
        • Final simplified expression: x2+14x+49x^2 + 14x + 49
    • Problem 4: Expand and simplify (x+2)2(x + 2)^2
      • Rewrite as repeated multiplication: (x+2)(x+2)(x + 2)(x + 2)
      • Apply the identity with a=xa = x and b=2b = 2:
        • a2=x2a^2 = x^2
        • 2ab=2×x×2=4x2ab = 2 \times x \times 2 = 4x
        • b2=22=4b^2 = 2^2 = 4
      • Final simplified expression: x2+4x+4x^2 + 4x + 4

  • Expansion of Triple Binomials (Cubic Polynomials):

    • Problem 6: Expand and simplify (x−4)(x+1)(x+1)(x - 4)(x + 1)(x + 1)
      • Step 1: Expand the repeated pair of identical binomial factors (x+1)(x+1)(x + 1)(x + 1).
        • (x+1)(x+1)=x2+2x+1(x + 1)(x + 1) = x^2 + 2x + 1
      • Step 2: Multiply the remaining linear binomial (x−4)(x - 4) by the quadratic trinomial (x2+2x+1)(x^2 + 2x + 1).
        • (x−4)(x2+2x+1)=x(x2+2x+1)−4(x2+2x+1)(x - 4)(x^2 + 2x + 1) = x(x^2 + 2x + 1) - 4(x^2 + 2x + 1)
      • Step 3: Distribute xx through the trinomial:
        • x×x2=x3x \times x^2 = x^3
        • x×2x=2x2x \times 2x = 2x^2
        • x×1=xx \times 1 = x
        • Result: x3+2x2+xx^3 + 2x^2 + x
      • Step 4: Distribute −4-4 through the trinomial:
        • −4×x2=−4x2-4 \times x^2 = -4x^2
        • −4×2x=−8x-4 \times 2x = -8x
        • −4×1=−4-4 \times 1 = -4
        • Result: −4x2−8x−4-4x^2 - 8x - 4
      • Step 5: Combine all terms together:
        • x3+2x2+x−4x2−8x−4x^3 + 2x^2 + x - 4x^2 - 8x - 4
      • Step 6: Group and combine like terms:
        • Cubic term: x3x^3
        • Quadratic terms: 2x2−4x2=−2x22x^2 - 4x^2 = -2x^2
        • Linear terms: x−8x=−7xx - 8x = -7x
        • Constant term: −4-4
      • Final simplified expression: x3−2x2−7x−4x^3 - 2x^2 - 7x - 4

Factorisation of Algebraic Expressions

  • Foundational Principles of Factorisation:

    • Factorisation is the reverse process of expansion, transforming a polynomial into a product of simpler polynomials or linear factors.

  • Factoring by Extracting a Common Monomial Factor:

    • Method: Identify the greatest common divisor (highest common factor) of all numerical coefficients and the lowest power of any shared variables, factor it out, and divide each term by this factor.
    • Problem 1: Factorise 5y2−7y5y^2 - 7y
      • Analyze terms: 5y25y^2 and −7y-7y.
      • Numerical coefficients 55 and 77 share no common factors other than 11.
      • Variable terms y2y^2 and yy share the common factor yy.
      • Factoring out yy:
        • 5y2y=5y\frac{5y^2}{y} = 5y
        • −7yy=−7\frac{-7y}{y} = -7
      • Factorised result: y(5y−7)y(5y - 7)

  • Factoring Monic Quadratic Trinomials (x2+bx+cx^2 + bx + c):

    • Method (Sum and Product Method):
      • To factor x2+bx+cx^2 + bx + c into (x+p)(x+q)(x + p)(x + q), identify two integers pp and qq such that:
        • p×q=cp \times q = c (Product of the factors equals the constant term)
        • p+q=bp + q = b (Sum of the factors equals the linear coefficient)
    • Problem 2: Factorise x2+10x+21x^2 + 10x + 21
      • Identify coefficients: b=10b = 10, c=21c = 21.
      • Find factors of 2121 that sum to 1010:
        • 1×21=21  ⟹  1+21=221 \times 21 = 21 \implies 1 + 21 = 22
        • 3×7=21  ⟹  3+7=103 \times 7 = 21 \implies 3 + 7 = 10
      • Integers are 33 and 77.
      • Factorised result: (x+7)(x+3)(x + 7)(x + 3)
    • Problem 3: Factorise x2+6x−16x^2 + 6x - 16
      • Identify coefficients: b=6b = 6, c=−16c = -16.
      • Find factors of −16-16 that sum to 66:
        • 16×(−1)=−16  ⟹  16+(−1)=1516 \times (-1) = -16 \implies 16 + (-1) = 15
        • 4×(−4)=−16  ⟹  4+(−4)=04 \times (-4) = -16 \implies 4 + (-4) = 0
        • 8×(−2)=−16  ⟹  8+(−2)=68 \times (-2) = -16 \implies 8 + (-2) = 6
      • Integers are 88 and −2-2.
      • Factorised result: (x+8)(x−2)(x + 8)(x - 2)
    • Problem 4: Factorise x2−4x+4x^2 - 4x + 4
      • Identify coefficients: b=−4b = -4, c=4c = 4.
      • Find factors of 44 that sum to −4-4:
        • (−1)×(−4)=4  ⟹  (−1)+(−4)=−5(-1) \times (-4) = 4 \implies (-1) + (-4) = -5
        • (−2)×(−2)=4  ⟹  (−2)+(−2)=−4(-2) \times (-2) = 4 \implies (-2) + (-2) = -4
      • Integers are −2-2 and −2-2.
      • Factorised result: (x−2)(x−2)(x - 2)(x - 2) or (x−2)2(x - 2)^2
    • Problem 7: Factorise x2+11x+28x^2 + 11x + 28
      • Identify coefficients: b=11b = 11, c=28c = 28.
      • Find factors of 2828 that sum to 1111:
        • 1×28=28  ⟹  1+28=291 \times 28 = 28 \implies 1 + 28 = 29
        • 2×14=28  ⟹  2+14=162 \times 14 = 28 \implies 2 + 14 = 16
        • 4×7=28  ⟹  4+7=114 \times 7 = 28 \implies 4 + 7 = 11
      • Integers are 77 and 44.
      • Factorised result: (x+7)(x+4)(x + 7)(x + 4)
    • Problem 8: Factorise x2+2x−35x^2 + 2x - 35
      • Identify coefficients: b=2b = 2, c=−35c = -35.
      • Find factors of −35-35 that sum to 22:
        • 35×(−1)=−35  ⟹  35+(−1)=3435 \times (-1) = -35 \implies 35 + (-1) = 34
        • 7×(−5)=−35  ⟹  7+(−5)=27 \times (-5) = -35 \implies 7 + (-5) = 2
      • Integers are 77 and −5-5.
      • Factorised result: (x+7)(x−5)(x + 7)(x - 5)
    • Supplementary Example: Factorise x2+7x+10x^2 + 7x + 10
      • Identify coefficients: b=7b = 7, c=10c = 10.
      • Find factors of 1010 that sum to 77:
        • 1×10=10  ⟹  1+10=111 \times 10 = 10 \implies 1 + 10 = 11
        • 5×2=10  ⟹  5+2=75 \times 2 = 10 \implies 5 + 2 = 7
      • Integers are 55 and 22.
      • Factorised result: (x+5)(x+2)(x + 5)(x + 2)

  • Factoring by Difference of Two Squares:

    • Theoretical formula: a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b).
    • Problem 5: Factorise 16−x216 - x^2
      • Recognize terms as perfect squares:
        • 16=4216 = 4^2
        • x2=(x)2x^2 = (x)^2
      • Apply difference of two squares identity with a=4a = 4 and b=xb = x:
      • Factorised result: (4+x)(4−x)(4 + x)(4 - x)
    • Problem 6: Factorise 4x2−94x^2 - 9
      • Recognize terms as perfect squares:
        • 4x2=(2x)24x^2 = (2x)^2
        • 9=329 = 3^2
      • Apply difference of two squares identity with a=2xa = 2x and b=3b = 3:
      • Factorised result: (2x+3)(2x−3)(2x + 3)(2x - 3)

Rearranging Formulas (Changing the Subject of the Formula)

  • Principles of Changing the Subject:

    • To make a variable the subject of a formula, perform inverse mathematical operations systematically to isolate the specified variable on one side of the equation with a coefficient of +1+1.

  • Linear Rearrangement:

    • Problem 7: Make xx the subject of the formula n=4x+pn = 4x + p
      • Step 1: Isolate the term containing xx (4x4x) by subtracting pp from both sides of the equation:
        • n−p=4xn - p = 4x
      • Step 2: Isolate xx by dividing both sides by the numerical coefficient 44:
        • n−p4=x\frac{n - p}{4} = x
      • Result:
        • x=n−p4x = \frac{n - p}{4}

  • Radical Equations (Involving Square Roots):

    • Problem 8: Make xx the subject of the formula p=x+3p = \sqrt{x + 3}
      • Step 1: Eliminate the radical (square root) by squaring both sides of the equation:
        • p2=(x+3)2p^2 = (\sqrt{x + 3})^2
        • p2=x+3p^2 = x + 3
      • Step 2: Isolate xx by subtracting 33 from both sides of the equation:
        • p2−3=xp^2 - 3 = x
      • Result:
        • x=p2−3x = p^2 - 3

  • Rearrangement Requiring Factorisation:

    • Problem 9: Make xx the subject of the formula dx=cx+qdx = cx + q
      • Step 1: Group all terms containing the target subject variable xx on one side of the equation by subtracting cxcx from both sides:
        • dx−cx=qdx - cx = q
      • Step 2: Factorise out the common variable factor xx on the left-hand side:
        • x(d−c)=qx(d - c) = q
      • Step 3: Divide both sides by the algebraic binomial factor (d−c)(d - c) to isolate xx:
        • x=qd−cx = \frac{q}{d - c}