FACTORING TRINOMIALS BY SPLITTING THE MIDDLE TERM

Factor each trinomial by splitting the middle term. 

Example 1 :

(p - q)² - 6(p - q) - 16

Solution :

=  (p - q)² - 6(p - q) - 16

Let t = p - q

  =  t² - 6t - 16

  =  t² - 8t + 2t - 16

  =  t(t - 8) + 2(t - 8)

  =  (t + 2)(t - 8)

  =  (p - q + 2)(p - q - 8)

The factors are (p - q + 2) and (p - q - 8).

Example 2 :

m² + 2mn - 24n²

Solution :

  =  m² + 2mn - 24n²

  =  m² + 6mn - 4mn - 24n²

  =  m(m + 6n) - 4n (m + 6n)

  =  (m - 4n)(m + 6n)

The factors are (m - 4n) and (m + 6n).

Example 3 :

 √5 a² + 2a - 3√5

Solution :

  =  √5 a² + 2a - 3√5

√5(3√5)  =  15

  =  √5 a² + 5a - 3a - 3√5

  =  √5 a(a + √5) - 3(a + √5)

  =  (√5a - 3)(a + √5)

The factors are (√5a - 3) and (a + √5).

Example 4 : 

a⁴ - 3a² + 2

Solution :

  =  a⁴ - 3a² + 2

  =  (a²)² - 3a² + 2

Let t = a²

  =  t² - 3t + 2

  =  t² - 2t - t + 2

  =  t(t - 2) - 1(t - 2)

  =  (t - 1)(t - 2)

Substitute a² for t.

  =  (a² - 1)(a² - 2)

  =  (a + 1)(a - 1)(a² - 2)

The factors are (a + 1), (a - 1) and (a² - 2).

Example 5 : 

8m³ - 2m²n - 15mn^2.

Solution :

  =  8m³ - 2m²n - 15mn²

  =  m(8m² - 2mn - 15n²)

  =  m(8m² - 12mn + 10mn - 15n²)

  =  m[4m(2m - 3n) + 5n(2m - 3n)]

  =  m(4m + 5n)(2m - 3n)

The factors are m, (4m + 5n) and (2m - 3n).

Example 6 :

(1/x²) + (1/y²) + (2/xy)

Solution :

  =  (1/x²) + (1/y²) + (2/xy)

  =  (1/x)² + (1/y)² + 2(1/x) (1/y)

  =  [(1/x) + (1/y)]²

=  (1/x + 1/y)(1/x + 1/y)

The factors are (1/x + 1/y) and (1/x + 1/y).

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