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:
- Factor the first term: \(x^2 = x \cdot x\).
- Factor the last term: Choose factors that multiply to obtain the last term and subtract to obtain the middle term.
- Determine the signs: by adding or subtracting the product of the inner and outer terms.
- Check by multiplying.
Example:
Factor the trinomial \(x^2 - 3x - 10\).
- Factor the first term: \(x^2 = x \cdot x\).
- Factor the last term: \(-10 = -5 \cdot 2\) and \(-5 + 2 = -3\), which is our middle term.
- Determine the signs: One sign is positive and the other is negative because \(-5 + 2 = -3\).
- 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:
- \(x^2 - 5x - 6\)
- \(x^2 - 7x + 10\)
- \(x^2 - 6x - 16\)
- \(x^2 - 8x - 9\)
- \(x^2 - 3x - 10\)
- \(x^2 - 4x - 21\)
- \(x^2 - 5x - 14\)
- \(x^2 - 6x - 16\)
- \(x^2 - 7x - 18\)
- \(x^2 - 9x - 22\)
Solutions:
- \(x^2 - 5x - 6 = (x - 6)(x + 1)\)
- \(x^2 - 7x + 10 = (x - 2)(x - 5)\)
- \(x^2 - 6x - 16 = (x - 8)(x + 2)\)
- \(x^2 - 8x - 9 = (x - 9)(x + 1)\)
- \(x^2 - 3x - 10 = (x - 5)(x + 2)\)
- \(x^2 - 4x - 21 = (x - 7)(x + 3)\)
- \(x^2 - 5x - 14 = (x - 7)(x + 2)\)
- \(x^2 - 6x - 16 = (x - 8)(x + 2)\)
- \(x^2 - 7x - 18 = (x - 9)(x + 2)\)
- \(x^2 - 9x - 22 = (x - 11)(x + 2)\)
Remember to always check your results by multiplying the factors. If the product matches the original trinomial, then the factoring is correct.
