## 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.