Could you explain how to factor the following trinomials : x2 + bx + c and ax2 + bx + c.
Is there more than one way to factor this? Could you answer this question using both words and mathematical notation.
Sure! I'd be happy to explain how to factor trinomials of the form x^2 + bx + c and ax^2 + bx + c.
To factor the trinomial x^2 + bx + c, you're looking for two binomials that multiply together to give you x^2 + bx + c. The general form of the factored trinomial is (x + p)(x + q), where p and q are the values you're looking for.
To find p and q, you need to consider the constant term c and the coefficient of the linear term b. First, find two numbers that multiply together to give you c, and also add up to give you b. These two numbers will be p and q.
For example, if you have the trinomial x^2 + 5x + 6, you want to find two numbers that multiply to give you 6 and add up to give you 5. In this case, the numbers are 3 and 2. So, the factored form of the trinomial is (x + 3)(x + 2).
Now let's move on to factoring the trinomial ax^2 + bx + c, where a is not equal to 1. In this case, you'll still look for two binomials, but they'll be of the form (mx + p)(nx + q), where m and n are the numbers you need to find.
To factor the trinomial ax^2 + bx + c, you can use a method called "AC method" or "product-sum method".
First, multiply the coefficient of the quadratic term a by the constant term c. Then, find two numbers that multiply together to give you ac, and also add up to give you b. These two numbers will be p and q.
For example, if you have the trinomial 2x^2 + 7x + 3, multiply 2 and 3 to get 6. Then, find two numbers that multiply together to give you 6 and add up to give you 7. In this case, the numbers are 6 and 1. So, the factored form of the trinomial is (2x + 1)(x + 3).
In summary, to factor trinomials of the form x^2 + bx + c and ax^2 + bx + c, you can use the method of finding two numbers that multiply together to give you the constant term c and add up to give you the coefficient of the linear term b. However, for the trinomial ax^2 + bx + c, where a is not equal to 1, the AC method is used to find the values of m, n, p, and q that form the factored form (mx + p)(nx + q).
I hope this explanation helps! Let me know if you have any further questions.