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.
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To factor the trinomial forms x^2 + bx + c and ax^2 + bx + c, there are a few different methods you can use.
Method 1: Factoring by Grouping
1. Write down the trinomial in the form ax^2 + bx + c.
2. Look for two numbers, let's call them m and n, that add up to b and multiply to c.
3. Split the middle term of the trinomial using m and n. This means rewriting bx as mx + nx.
4. Group the terms so that you have two pairs: (ax^2 + mx) + (nx + c).
5. Factor out the common factor from each pair. This will give you a common binomial factor: x(ax + m) + (ax + m).
6. Factor out the common binomial factor from both pairs: (ax + m)(x + n).
7. Simplify if possible.
Method 2: Factoring using the Quadratic Formula
1. Write down the trinomial in the form ax^2 + bx + c.
2. Identify the values of a, b, and c from the trinomial.
3. Use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
4. Substitute the values of a, b, and c into the quadratic formula.
5. Simplify and solve for the values of x.
6. Rewrite x = (-b ± √(b^2 - 4ac)) / (2a) as two separate equations.
7. These equations represent the factors of the trinomial.
Method 3: Factoring by Trial and Error
1. Write down the trinomial in the form ax^2 + bx + c.
2. Look for two numbers, let's call them m and n, that add up to b and multiply to ac.
3. Write down all possible factor pairs of ac.
4. Find the pair that adds up to b.
5. Rewrite the middle term using the pair of numbers, giving you ax^2 + mx + nx + c.
6. Group the terms into pairs: (ax^2 + mx) + (nx + c).
7. Factor out the common factor from each pair: x(ax + m) + n(ax + m).
8. Factor out the common binomial factor from both pairs: (ax + m)(x + n).
9. Simplify if possible.
By using these methods, you can factor trinomials of the form x^2 + bx + c and ax^2 + bx + c. There may be cases where one method is easier or more applicable than the others, so it's good to be familiar with multiple factoring techniques.
To factor the trinomial form x^2 + bx + c, you need to find two binomials that, when multiplied, give you the original trinomial.
Step 1: Identify the factors of the constant term (c).
First, we need to identify the factors of the constant term (c). These factors are the possible values for the coefficients of the binomials. For example, if the constant term is 10, the factors could be 1 and 10 or 2 and 5.
Step 2: Determine the binomial form.
Next, we need to determine the form of the binomials. Since x^2 is the first term, the binomials will have the form (x + ?)(x + ?). The question marks represent the coefficients we need to find.
Step 3: Find the coefficients.
To find the coefficients, we need to inspect the middle term (bx). The sum of the coefficients in the binomials should equal b. For example, if the middle term is 4x, the coefficients could be 2 and 2, where 2 + 2 = 4. So the binomial form would be (x + 2)(x + 2).
Let's try an example:
Factor the trinomial x^2 + 5x + 6.
1. Identify the factors of the constant term (6): 1 and 6, 2 and 3.
2. Determine the binomial form: (x + ?)(x + ?).
3. Find the coefficients: The sum of the coefficients should equal 5.
- We can use the factors 2 and 3, where 2 + 3 = 5.
- So the binomial form is (x + 2)(x + 3).
The factored form of x^2 + 5x + 6 is (x + 2)(x + 3).
Now let's move on to factoring the trinomial form ax^2 + bx + c.
The process for factoring ax^2 + bx + c is similar to that of x^2 + bx + c, with the addition of the coefficient a. The general form of the binomials will be (ax + ?)(x + ?).
Here's an example:
Factor the trinomial 2x^2 + 7x + 3.
1. Identify the factors of the constant term (3): 1 and 3.
2. Determine the binomial form: (2x + ?)(x + ?).
3. Find the coefficients: The sum of the coefficients should equal 7.
- We can use the factors 1 and 3, where 1 + 3 = 4.
- So the binomial form is (2x + 1)(x + 3).
The factored form of 2x^2 + 7x + 3 is (2x + 1)(x + 3).
Yes, there is only one way to factor these trinomials in order to get the correct factored form. The process involves finding the factors of the constant term and determining the coefficients in the binomials that sum up to the coefficient of the middle term.