Explain how to factor the following trinomials forms: x2 + bx + c and ax2 + bx + c. Is there more than one way to factor this? Show your answer using both words and mathematical notation.
Factor x^2 + 5x + 6
C = 6 = 1*6 = 2*3.
Select the pair of factors whose sum equals B which is 5.
We select 2 and 3, because their sum = 5.
Solution: (x+2)(x+3).
The solution can be checked by performing the multiplication and
combining like-terms.
Factor 2x^2 + 11x + 12.
This and the previous example can be done by trial and error, but the process can be tedious and very frustrating.
We'll use the AC Method:
A*C = 2 * 12 = 24 = 3*8 = 4*3.
We select the pair of factors whose sum = B(11). Therfore, we select 3 and 8.
2x^2 + (3x+8x) + 12
Arrange the 4 terms into two factorable binomials:
(2x^2+3x) + (8x+12)
x(2x+3) + 4(2x+3)
(2x+3)(x+4).
I hope this helps.
To factor the trinomial forms x^2 + bx + c and ax^2 + bx + c, we'll need to find two binomials whose product gives us the original trinomial.
1. Factoring x^2 + bx + c:
- First, we need to identify two numbers that add up to b and multiply to c.
- Let's call these numbers m and n.
- Now, we can rewrite the trinomial as x^2 + mx + nx + c.
- Next, we group the terms into two pairs: (x^2 + mx) and (nx + c).
- Within each pair, we factor out the common factor, resulting in: x(x + m) + n(x + m).
- Notice that both terms in the parentheses are now the same.
- Now, we can factor out this common binomial, giving us: (x + n)(x + m).
- So, the factored form of x^2 + bx + c is (x + n)(x + m).
2. Factoring ax^2 + bx + c:
- Starting with the trinomial ax^2 + bx + c, we aim to find two numbers that multiply to a*c (product of the first and last coefficients) and add up to b (the middle coefficient).
- Once again, let's call these numbers m and n.
- We rewrite the trinomial as ax^2 + mx + nx + c.
- Just like before, we group the terms into two pairs: (ax^2 + mx) and (nx + c).
- Within each pair, we factor out the common factor, resulting in: x(ax + m) + n(ax + m).
- Notice that both terms in the parentheses are now the same.
- We can now factor out this common binomial, giving us: (ax + n)(x + m).
- So, the factored form of ax^2 + bx + c is (ax + n)(x + m).
So, to summarize:
- For x^2 + bx + c, the factored form is (x + n)(x + m).
- For ax^2 + bx + c, the factored form is (ax + n)(x + m).
Note: There may be cases where the trinomial cannot be factored using integer values, in which case, factoring may involve more complex methods such as completing the square or using the quadratic formula.