Scanned Document

Perfect Square Trinomial Formula

  • Definition: Used to factor trinomials into a factor that is squared.

  • Conditions to Use:

    • First term must be a perfect square: (x)².

    • Second term must also be a perfect square: (y)².

    • Middle term must be obtained by multiplying xy and doubling it: 2xy.

  • Formulas:

    • x² + 2xy + y² = (x + y)²

    • x² - 2xy + y² = (x - y)²

  • Example: Factor a² + 6a + 9:

    • Rewritten as: a² + 2•a•3 + 3²

    • Factors to: (a + 3)²

Example of Factoring Perfect Square Trinomials

  • Example: Factor 4m² - 20mn + 25n²:

    • 4m² = (2m)²

    • 20mn is from: 2•2m•5n

    • 25n² = (5n)²

    • Therefore, 4m² - 20mn + 25n² = (2m - 5n)²

  • Another Example: Factor 16x⁶ + 40x³y² + 25y¹⁴:

    • 16x⁶ = (4x³)²

    • 40x³y² from 2•4x³•5y

    • 25y¹⁴ = (5y)²

    • Result: (4x³ + 5y)²

Sum of Two Cubes and Difference of Two Cubes Formulas

  • Formulas for Sums:

    • x³ + y³ = (x + y)(x² - xy + y²)

  • Formulas for Differences:

    • x³ - y³ = (x - y)(x² + xy + y²)

  • Conditions to Use:

    • Check if both terms are perfect cubes.

  • Example: Factor a³ + 8b³:

    • a³ = (a)³, 8b³ = (2b)³

    • Result: (a + 2b)((a)² - a•2b + (2b)²) = (a + 2b)(a² - 2ab + 4b²)

Example of Factoring Cubes

  • Example: Factor 125m²n¹² - 125m³n¹²:

    • Rewrite 125 as 5³

    • Expression becomes: (5)³ - (m³n)³

    • Result: (5 - m³n)[(5)² + 5•m³n + (m³n)²]

  • Another Example: Factor 27x³y¹² - 64:

    • 27x³y¹² = (3xy)³, 64 = (4)³

    • Result: (3xy - 4)[(3xy)² + (3xy)(4) + (4)²] = (3xy - 4)(9x²y² + 12xy + 16)

Combining Different Methods of Factoring

  • Example: Factor 18ax² - 32a:

    • Identify GCF: 2a

    • Factor out GCF: 2a(9x² - 16)

    • Apply Difference of Two Squares: 2a[(3x)² - (4)²] = 2a(3x + 4)(3x - 4)

  • Another Example: Factor x⁴ - 81 + 6x³ - 54x:

    • Group terms: (x⁴ - 81) + (6x³ - 54x)

    • Apply Difference of Two Squares to first part: (x² - 9)(x² + 9)

    • Use GCF on second part: 6x(x² - 9)

    • Combined result: (x² - 9)(x² + 6x + 9) = (x - 3)(x + 3)(x + 3)²

Check Your Knowledge

  • Exercises (to factor each trinomial):

    1. m² - 121

    2. 9x² + 6xy + y²

    3. 64 - 9366

    4. 25p² - 15pq + 9q²

    5. 25₤16 - 36936

    6. 27a + 646³ C15

    7. 121pq⁰ - 66p²q³r + 9r⁸

    8. 4m²n³ - 36n²p⁸

    9. 8a⁸b⁴ - 40a⁵b⁶ + 50a²b⁸

    10. 16a663 + 54618c³

Answers

  • Answers to exercises:

    1. (m + 11)(m - 11)

    2. (3x + y)²

    3. (4ab²)(16 + 4ab² + a²b⁴)

    4. N/A

    5. (5f⁸ - 6918)(5f³ + 6918)

    6. (3a² + 4bc⁵)(9a - 12a²bc⁵ + 16c²)

    7. (11p²q⁵ - 3r⁴)²

    8. An²(mn³ - 3p²)(mn³ + 3p²)

    9. 2a²b⁴(2a³ - 5²)²

    10. 263(2a² + 36c)(4a⁴ - 6a²b⁵c + 9610c²)