Can you explain how to factor the following trinomials forms: x2 + bx + c and ax2 + bx + c. Can there be more than one way to factor a problem like this? can you please show it in written word and step by step?
Your question usually requires as least one chapter in a math text book.
here are some pages you might want to look at.
http://www.themathpage.com/alg/factoring-trinomials.htm
http://www.algebrahelp.com/lessons/factoring/trinomial/pg3.htm
Salman Khan has two good videos:
http://www.youtube.com/watch?v=nOZTe8jU2g4
http://www.youtube.com/watch?v=ISPxJ6JXT8o&feature=relmfu
Certainly! I'd be happy to explain how to factor trinomials of the form x^2 + bx + c and ax^2 + bx + c.
1. Factoring trinomials of the form x^2 + bx + c:
To factor trinomials in this form, we look for two numbers that multiply together to give us the constant term (c), and add up to give us the coefficient of the linear term (b).
Let's take an example: x^2 + 5x + 6
First, we need to find two numbers that multiply to give us 6, the constant term, and add up to give us 5, the coefficient of the linear term. In this example, the numbers are 2 and 3, as 2 * 3 = 6 and 2 + 3 = 5.
Now, we rewrite the trinomial using these two numbers:
x^2 + 2x + 3x + 6
Now, we group the terms and factor them by pairs:
(x^2 + 2x) + (3x + 6)
Now, factor out the greatest common factor from each pair:
x(x + 2) + 3(x + 2)
Notice that (x + 2) is common to both terms. We factor it out:
(x + 2)(x + 3)
And that's the factored form of the trinomial.
2. Factoring trinomials of the form ax^2 + bx + c:
Factoring trinomials in this form is similar to the previous case, but we have an additional challenge of finding two numbers that multiply to give us ac (the product of a and c), and also add up to give us b, the coefficient of the linear term.
Let's take an example: 2x^2 + 5x + 3
Again, we need to find two numbers that multiply to give us 6 (2 * 3), the product of a and c, and add up to 5, the coefficient of the linear term. In this example, the numbers are 2 and 3.
Now, we rewrite the trinomial using these two numbers:
2x^2 + 2x + 3x + 3
Next, we group the terms and factor them by pairs:
(2x^2 + 2x) + (3x + 3)
Now, factor out the greatest common factor from each pair:
2x(x + 1) + 3(x + 1)
Again, notice that (x + 1) is common to both terms. We factor it out:
(x + 1)(2x + 3)
So, the factored form of the trinomial is (x + 1)(2x + 3).
Regarding your question about multiple ways to factor these trinomials, in most cases, there will be only one correct way to factor them. However, there might be different ways to rearrange the terms or choose the grouping technique, which could lead to different but equivalent factored forms.
I hope this step-by-step explanation helps you understand how to factor both x^2 + bx + c and ax^2 + bx + c trinomials! Let me know if you have any further questions.