Factoring Lesson C by ChatGPT-4

Factoring Trinomials of the Form \(x^2 - bx - c\)

Factoring trinomials, which are polynomials with three terms, can initially seem challenging. However, with practice, it becomes routine. If a trinomial factors, it will factor into the product of two binomials.

Steps to Factor Trinomials:

  1. Factor the first term: \(x^2 = x \cdot x\).
  2. Factor the last term: Choose factors that multiply to obtain the last term and subtract to obtain the middle term.
  3. Determine the signs: by adding or subtracting the product of the inner and outer terms.
  4. Check by multiplying.

Example:

Factor the trinomial \(x^2 - 3x - 10\).

  1. Factor the first term: \(x^2 = x \cdot x\).
  2. Factor the last term: \(-10 = -5 \cdot 2\) and \(-5 + 2 = -3\), which is our middle term.
  3. Determine the signs: One sign is positive and the other is negative because \(-5 + 2 = -3\).
  4. Check by multiplying: \((x - 5)(x + 2) = x^2 - 3x - 10\).

So, \(x^2 - 3x - 10 = (x - 5)(x + 2)\).

Exercises:

Factor the following trinomials:

  1. \(x^2 - 5x - 6\)
  2. \(x^2 - 7x + 10\)
  3. \(x^2 - 6x - 16\)
  4. \(x^2 - 8x - 9\)
  5. \(x^2 - 3x - 10\)
  6. \(x^2 - 4x - 21\)
  7. \(x^2 - 5x - 14\)
  8. \(x^2 - 6x - 16\)
  9. \(x^2 - 7x - 18\)
  10. \(x^2 - 9x - 22\)

Solutions:

  1. \(x^2 - 5x - 6 = (x - 6)(x + 1)\)
  2. \(x^2 - 7x + 10 = (x - 2)(x - 5)\)
  3. \(x^2 - 6x - 16 = (x - 8)(x + 2)\)
  4. \(x^2 - 8x - 9 = (x - 9)(x + 1)\)
  5. \(x^2 - 3x - 10 = (x - 5)(x + 2)\)
  6. \(x^2 - 4x - 21 = (x - 7)(x + 3)\)
  7. \(x^2 - 5x - 14 = (x - 7)(x + 2)\)
  8. \(x^2 - 6x - 16 = (x - 8)(x + 2)\)
  9. \(x^2 - 7x - 18 = (x - 9)(x + 2)\)
  10. \(x^2 - 9x - 22 = (x - 11)(x + 2)\)

Remember to always check your results by multiplying the factors. If the product matches the original trinomial, then the factoring is correct.

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