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.
Let's factor the trinomial x² + bx + c. We know that if b²-4c >= 0 then this trinomial has zeros and it can be factorized. It's zeros are
x1 = [ -b + √(b²-4c) ] / 2
and
x2 = [ -b - √(b²-4c) ] / 2.
So, we can factor it as
x² + bx + c = (x-x1)(x-x2).
In a case of trinomial ax² + bx + c we can factorize it as
ax² + bx + c = a(x² + b/a·x + c/a),
where the trinomial x² + b/a·x + c/a has the same form as x² + bx + c.
To factor the trinomial forms x² + bx + c and ax² + bx + c, there are a few strategies you can use. Let's break down each form separately.
1. Factoring x² + bx + c:
- Look for two numbers that multiply to give the constant term c and add up to give the coefficient of x, which is b.
- Write the factors of c as pairs and test each pair until you find the one that satisfies the conditions mentioned above.
- Once you have found the correct pair of factors, rewrite the middle term bx as the sum of the two numbers.
- Now you can factor the trinomial by grouping or by using special factoring formulas (if applicable).
For example, let's say we need to factor the trinomial x² + 5x + 6:
- The constant term c is 6, and we need to find two numbers that multiply to give 6 and add up to give 5 (coefficient of x).
- The factors of 6 are (1, 6) and (2, 3).
- Testing the pairs, we find that 2 and 3 satisfy the conditions, as 2 + 3 = 5 and 2 * 3 = 6.
- Rewrite the middle term: x² + 2x + 3x + 6.
- Now you can factor by grouping: (x² + 2x) + (3x + 6).
- Factor out common terms from each group: x(x + 2) + 3(x + 2).
- Notice that (x + 2) is common to both terms, so we can further simplify: (x + 2)(x + 3).
- And there you have it, the factored form of x² + 5x + 6 is (x + 2)(x + 3).
2. Factoring ax² + bx + c:
- This form is similar to the previous one, but now there is an extra coefficient a in front of x².
- The best approach is to first factor out the common factor a, if it exists.
- Then, you can use the same methods as before for factoring the trinomial x² + bx + c.
For example, let's say we need to factor the trinomial 2x² + 7x + 3:
- Notice that there is a common factor of 1 (which can be ignored) and the leading coefficient is 2.
- Divide the entire trinomial by 2 (factor out the common factor): (2x² + 7x + 3) ÷ 2 = x² + 3.5x + 1.5.
- Now, we can factor the simplified trinomial x² + 3.5x + 1.5 using the same process described earlier.
- Find two numbers that multiply to give 1.5 and add up to give 3.5.
- After testing, we find that the factors 1 and 1.5 meet the conditions.
- Rewrite the middle term: x² + x + 2.5x + 1.5.
- Factor by grouping: (x² + x) + (2.5x + 1.5).
- Factor out common terms from each group: x(x + 1) + 1.5(x + 1).
- Notice that (x + 1) is common to both terms, so we can further simplify: (x + 1)(x + 1.5).
- Final result after factoring out the common factor: 2x² + 7x + 3 = 2(x + 1)(x + 1.5).
In conclusion, there are multiple ways to factor trinomial expressions depending on the specific form and given coefficients. The key is to find suitable factor pairs for the constant term and then use appropriate factoring techniques to simplify the expression.