Literal Calculation: Comprehensive Guide to Expanding and Factorizing by Yvan Monka

Fundamental Definitions: Sums and Products

These notes are based on the curriculum by Yvan Monka from the Académie de Strasbourg (www.maths-et-tiques.fr). The material covers the essential principles of literal calculation, focusing on expanding (developing) and factorizing algebraic expressions.

Definitions and Core Concepts
  • Developing: To develop is to transform a mathematical product into a sum.
    • Example: x(4−y)=4x−xyx(4 - y) = 4x - xy
  • Factorizing: To factorize is to transform a mathematical sum into a product.
    • Example: 4x−xy=x(4−y)4x - xy = x(4 - y)
Identification of Expressions

Expressions are classified by their last operational step (the outer operation):

  • Sums (or differences) of terms:
    • x−3x - 3
    • (2x+4)+3x(2x + 4) + 3x
    • (5−x)−(9+9x)(5 - x) - (9 + 9x)
    • 3+(2+3x)(x−2)3 + (2 + 3x)(x - 2)
  • Products of factors:
    • (6x+1)×(x−1)(6x + 1) \times (x - 1)
    • 2×(1+6x)2 \times (1 + 6x)
    • (8−x)×(2+x)(8 - x) \times (2 + x)
    • (3+8x)×(x−8)(3 + 8x) \times (x - 8)
Video Resources
  • Overview of Developing: https://youtu.be/gSa851JJn6c
  • Overview of Factorizing: https://youtu.be/kQGWtMOHbrA
  • Sum vs Product: https://youtu.be/FTi9WOQsq3w

Expanding Expressions: Simple and Double Distributivity

1. Simple Distributivity

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

Example:6(x+5)=6x+306(x + 5) = 6x + 30 In this case, the product 6×(x+5)6 \times (x + 5) is transformed by multiplying 66 by each term inside the parentheses (6×56 \times 5 and 6×x6 \times x).

Methodology for expansion:

  • A=4(5+x)=4×5+4×x=20+4xA = 4(5 + x) = 4 \times 5 + 4 \times x = 20 + 4x
  • B=5(x−2)=5×x−5×2=5x−10B = 5(x - 2) = 5 \times x - 5 \times 2 = 5x - 10
  • C=(4x+6)×3=3×4x+3×6=12x+18C = (4x + 6) \times 3 = 3 \times 4x + 3 \times 6 = 12x + 18
  • D=−6(−2x+4)=−6×(−2x)+(−6)×4=12x−24D = -6(-2x + 4) = -6 \times (-2x) + (-6) \times 4 = 12x - 24
  • E=−x(2−3x)=−x×2−x×(−3x)=−2x+3x2E = -x(2 - 3x) = -x \times 2 - x \times (-3x) = -2x + 3x^2
  • F=−(5−x)=−5+xF = -(5 - x) = -5 + x

Crucial Rule: A minus sign (−-) placed in front of a pair of parentheses changes the signs of all terms within those parentheses: −(a−b)=−a+b-(a - b) = -a + b.

Video Links:

  • https://youtu.be/S_ckQpWzmG8
  • https://youtu.be/URNld8xsXgM
2. Double Distributivity

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

Example Case:(2+5x)(x+4)=2x+8+5x2+20x(2 + 5x)(x + 4) = 2x + 8 + 5x^2 + 20x

Method: Developing and Reducing Expressions:

  • Expression A:A=(2x+3)(x+8)A = (2x + 3)(x + 8)A=2x×x+2x×8+3×x+3×8A = 2x \times x + 2x \times 8 + 3 \times x + 3 \times 8A=2x2+16x+3x+24A = 2x^2 + 16x + 3x + 24A=2x2+19x+24A = 2x^2 + 19x + 24

  • Expression B:B=(−3+x)(4−5x)B = (-3 + x)(4 - 5x)B=−3×4−3×(−5x)+x×4+x×(−5x)B = -3 \times 4 - 3 \times (-5x) + x \times 4 + x \times (-5x)B=−12+15x+4x−5x2B = -12 + 15x + 4x - 5x^2B=−5x2+19x−12B = -5x^2 + 19x - 12

  • Expression C:C=2(3+x)(3−2x)C = 2(3 + x)(3 - 2x)C=2(3×3+3×(−2x)+x×3+x×(−2x))C = 2(3 \times 3 + 3 \times (-2x) + x \times 3 + x \times (-2x))C=2(9−6x+3x−2x2)C = 2(9 - 6x + 3x - 2x^2)C=2(−2x2−3x+9)C = 2(-2x^2 - 3x + 9)C=−4x2−6x+18C = -4x^2 - 6x + 18

  • Expression D:D=2x(1−x)−(x−3)(3x+2)D = 2x(1 - x) - (x - 3)(3x + 2)D=2x−2x2−(3x2+2x−9x−6)D = 2x - 2x^2 - (3x^2 + 2x - 9x - 6)D=2x−2x2−3x2−2x+9x+6D = 2x - 2x^2 - 3x^2 - 2x + 9x + 6D=−5x2+9x+6D = -5x^2 + 9x + 6

Video Links for Double Distributivity:

  • https://youtu.be/1EPOmbvoAlU
  • https://youtu.be/YS-3JI_z2f0
  • https://youtu.be/o6qVMmA3oTQ

Factorization: Using Common Factors

To factorize an expression, one must identify a common factor in every term of the sum.

Level 1: Simple Common Factors
  • A=3.5x−4.2x+2.1xA = 3.5x - 4.2x + 2.1x     Identifying xx: A=x(3.5−4.2+2.1)=1.4xA = x(3.5 - 4.2 + 2.1) = 1.4x
  • B=4t−5tx+3tB = 4t - 5tx + 3t     Identifying tt: B=t(4−5x+3)=t(7−5x)B = t(4 - 5x + 3) = t(7 - 5x)
  • C=4x−4y+8C = 4x - 4y + 8     Identifying 44: C=4x−4y+4×2=4(x−y+2)C = 4x - 4y + 4 \times 2 = 4(x - y + 2)
  • D=x2+3x−5x2D = x^2 + 3x - 5x^2     Identifying xx: D=x×x+3x−5x×x=x(x+3−5x)=x(−4x+3)D = x \times x + 3x - 5x \times x = x(x + 3 - 5x) = x(-4x + 3)
  • E=3t+9u+3E = 3t + 9u + 3     Identifying 33: E=3t+3×3u+3×1=3(t+3u+1)E = 3t + 3 \times 3u + 3 \times 1 = 3(t + 3u + 1)
  • F=3x2−xF = 3x^2 - x     Identifying xx: F=3x×x−1x=x(3x−1)F = 3x \times x - 1x = x(3x - 1)

Video Link: https://youtu.be/r3AzqvgLcI8

Level 2: Parenthetical Common Factors

  • Expression A:A=3(2+3x)−(5+2x)(2+3x)A = 3(2 + 3x) - (5 + 2x)(2 + 3x)     Common Factor: (2+3x)(2 + 3x)A=(2+3x)(3−(5+2x))A = (2 + 3x)(3 - (5 + 2x))A=(2+3x)(3−5−2x)=(2+3x)(−2−2x)A = (2 + 3x)(3 - 5 - 2x) = (2 + 3x)(-2 - 2x)

  • Expression B:B=(2−5x)2−(2−5x)(1+x)B = (2 - 5x)^2 - (2 - 5x)(1 + x)B=(2−5x)(2−5x)−(2−5x)(1+x)B = (2 - 5x)(2 - 5x) - (2 - 5x)(1 + x)     Common Factor: (2−5x)(2 - 5x)B=(2−5x)((2−5x)−(1+x))B = (2 - 5x)((2 - 5x) - (1 + x))B=(2−5x)(2−5x−1−x)=(2−5x)(1−6x)B = (2 - 5x)(2 - 5x - 1 - x) = (2 - 5x)(1 - 6x)

Level 3: Writing Modifications to Find Factors

Sometimes the common factor is not immediately visible.

  • Expression C:C=5(1−2x)−(4+3x)(2x−1)C = 5(1 - 2x) - (4 + 3x)(2x - 1)     Notice that (2x−1)(2x - 1) is the negative of (1−2x)(1 - 2x). We can rewrite the term to reveal the factor:     C=5(1−2x)+(4+3x)(1−2x)C = 5(1 - 2x) + (4 + 3x)(1 - 2x)C=(1−2x)(5+(4+3x))=(1−2x)(9+3x)C = (1 - 2x)(5 + (4+3x)) = (1 - 2x)(9 + 3x)

Video Link: https://youtu.be/UGTFELhE9Dw

Remarkable Identities

Remarkable identities are standard formulas used to expand or factorize expressions quickly.

1. Proprieties and Geometrical Interpretation
  1. First Identity: (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2
  2. Second Identity: (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2
  3. Third Identity: (a+b)(a−b)=a2−b2(a + b)(a - b) = a^2 - b^2

Geometric Note: The first identity (a+b)2(a + b)^2 can be visualized by considering the area of a square with side lengths (a+b)(a + b). The total area comprises a square with area a2a^2, a square with area b2b^2, and two rectangles with area abab, totaling a2+2ab+b2a^2 + 2ab + b^2.

Video Links:

  • Introduction to Identities: https://youtu.be/A8U1QVW7RaU
  • Geometric Proof: https://youtu.be/wDAdBXlZNK4
2. Expanding with Remarkable Identities

  • Basic Examples:

    • A=(x+3)2=x2+2×x×3+32=x2+6x+9A = (x + 3)^2 = x^2 + 2 \times x \times 3 + 3^2 = x^2 + 6x + 9
    • B=(3x−4)2=(3x)2−2×3x×4+42=9x2−24x+16B = (3x - 4)^2 = (3x)^2 - 2 \times 3x \times 4 + 4^2 = 9x^2 - 24x + 16
    • C=(x−3)(x+3)=x2−32=x2−9C = (x - 3)(x + 3) = x^2 - 3^2 = x^2 - 9
    • D=(2x+3)(2x−3)=(2x)2−32=4x2−9D = (2x + 3)(2x - 3) = (2x)^2 - 3^2 = 4x^2 - 9
    • E=(4−3x)(3x+4)=(4−3x)(4+3x)=42−(3x)2=16−9x2E = (4 - 3x)(3x + 4) = (4 - 3x)(4 + 3x) = 4^2 - (3x)^2 = 16 - 9x^2

  • Combining Identities with other expressions:

    • A=(2x−3)2+(x+5)(3−x)=4x2−12x+9+3x−x2+15−5x=3x2−14x+24A = (2x - 3)^2 + (x + 5)(3 - x) = 4x^2 - 12x + 9 + 3x - x^2 + 15 - 5x = 3x^2 - 14x + 24
    • B=(x−3)(x+3)−(4−3x)2=x2−9−(16−24x+9x2)=x2−9−16+24x−9x2=−8x2+24x−25B = (x - 3)(x + 3) - (4 - 3x)^2 = x^2 - 9 - (16 - 24x + 9x^2) = x^2 - 9 - 16 + 24x - 9x^2 = -8x^2 + 24x - 25
    • C=2(x+3)+(2x+3)(2x−3)=2x+6+4x2−9=4x2+2x−3C = 2(x + 3) + (2x + 3)(2x - 3) = 2x + 6 + 4x^2 - 9 = 4x^2 + 2x - 3

Expansion Videos:

  • Part 1: https://youtu.be/U98Tk89SJ5M
  • Part 2: https://youtu.be/7va96s4OfiM
3. Factorizing with Remarkable Identities

To factorize, identify the terms mapping to a2a^2, b2b^2, and (if applicable) 2ab2ab.

  • Basic Examples:

    • A=x2−2x+1=(x−1)2A = x^2 - 2x + 1 = (x - 1)^2 (a=x,b=1a=x, b=1)
    • B=4x2+12x+9=(2x+3)2B = 4x^2 + 12x + 9 = (2x + 3)^2 (a=2x,b=3a=2x, b=3)
    • C=9x2−4=(3x−2)(3x+2)C = 9x^2 - 4 = (3x - 2)(3x + 2) (a=3x,b=2a=3x, b=2, no 2ab2ab term)
    • D=25+16x2−40x=(5−4x)2D = 25 + 16x^2 - 40x = (5 - 4x)^2 (Rewritten: 25−40x+16x2,a=5,b=4x25 - 40x + 16x^2, a=5, b=4x)
    • E=1−49x2=(1−7x)(1+7x)E = 1 - 49x^2 = (1 - 7x)(1 + 7x) (a=1,b=7xa=1, b=7x)

  • Difference of Squares with Compound Terms:

    • Expression A:A=(2x+3)2−64A = (2x + 3)^2 - 64         Identified as a2−b2a^2 - b^2 with a=2x+3a = 2x + 3 and b=8b = 8.         A=((2x+3)−8)((2x+3)+8)A = ((2x + 3) - 8)((2x + 3) + 8)A=(2x+3−8)(2x+3+8)=(2x−5)(2x+11)A = (2x + 3 - 8)(2x + 3 + 8) = (2x - 5)(2x + 11)
    • Expression B:B=1−(2−5x)2B = 1 - (2 - 5x)^2         Identified as a2−b2a^2 - b^2 with a=1a = 1 and b=2−5xb = 2 - 5x.         B=(1−(2−5x))(1+(2−5x))B = (1 - (2 - 5x))(1 + (2 - 5x))B=(1−2+5x)(1+2−5x)=(−1+5x)(3−5x)B = (1 - 2 + 5x)(1 + 2 - 5x) = (-1 + 5x)(3 - 5x)

Factorization Videos:

  • Basic: https://youtu.be/T9T4IeYGEe4
  • Advanced: https://youtu.be/nLRRUMRyfZg and https://youtu.be/tO4p9TzMrls

Mentions légales

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