Factoring Lesson B 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 add 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 \(x^2 - 7x + 10\).

  1. Factor the first term: \(x^2 = x \cdot x\).
  2. Factor the last term: \(10 = 5 \cdot 2\) and \(5 + 2 = 7\), which is our middle term.
  3. Determine the signs: Both signs are negative because \(-5 - 2 = -7\).
  4. Check by multiplying: \((x - 5)(x - 2) = x^2 - 7x + 10\).

So, \(x^2 - 7x + 10 = (x - 5)(x - 2)\).

Exercises:

Factor the following trinomials:

  1. \(x^2 - 5x + 6\)
  2. \(x^2 - 9x + 20\)
  3. \(x^2 - 7x + 10\)
  4. \(x^2 - 11x + 30\)
  5. \(x^2 - 8x + 15\)
  6. \(x^2 - 10x + 21\)
  7. \(x^2 - 6x + 8\)
  8. \(x^2 - 12x + 35\)
  9. \(x^2 - 15x + 56\)
  10. \(x^2 - 13x + 40\)

Solutions:

  1. \(x^2 - 5x + 6 = (x - 3)(x - 2)\)
  2. \(x^2 - 9x + 20 = (x - 5)(x - 4)\)
  3. \(x^2 - 7x + 10 = (x - 5)(x - 2)\)
  4. \(x^2 - 11x + 30 = (x - 6)(x - 5)\)
  5. \(x^2 - 8x + 15 = (x - 5)(x - 3)\)
  6. \(x^2 - 10x + 21 = (x - 7)(x - 3)\)
  7. \(x^2 - 6x + 8 = (x - 4)(x - 2)\)
  8. \(x^2 - 12x + 35 = (x - 7)(x - 5)\)
  9. \(x^2 - 15x + 56 = (x - 8)(x - 7)\)
  10. \(x^2 - 13x + 40 = (x - 8)(x - 5)\)

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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