Explain how to factor the following trinomials forms: x2 + bx + c and ax2 + bx + c. I am totally lost still I read and read and the concept is lost in my head. Can someone please help me? I know there is more than one way to Factor this.
x^2 - 5^2 = (A - B)(x + 5) Could someone help me understand how to factor this.
Sure, I'd be happy to help you understand how to factor trinomials of the forms x^2 + bx + c and ax^2 + bx + c.
To factor trinomials, you need to find two binomials that, when multiplied, will give you the original trinomial. Let's start with the simpler form x^2 + bx + c.
1. Look at the constant term 'c': Find two numbers that multiply to 'c'.
2. Look at the coefficient of the 'x' term 'b': Find two numbers that add up to 'b'.
3. Write out the factors of 'c' and look for a pair whose sum matches the value of 'b'.
For example, if 'c' is 12 and 'b' is 7, you would look for two numbers whose product is 12 and whose sum is 7. The numbers in this case would be 3 and 4.
4. Rewrite the trinomial in factored form using the two numbers you found.
You would write x^2 + bx + c as (x + 4)(x + 3).
That's how you factor a trinomial of the form x^2 + bx + c. Now, let's move on to the more complex form ax^2 + bx + c.
To factor a trinomial of the form ax^2 + bx + c, where 'a' is not equal to 1:
1. Multiply 'a' and 'c' to get the product 'ac'.
2. Look for two numbers whose product is 'ac' and whose sum is 'b'.
3. Once you find those two numbers, use them to rewrite the middle term 'bx' as the sum or difference of these two numbers.
4. Group the terms and factor by grouping.
5. Factor out the greatest common factor from each group of terms.
6. Combine the factored groups to get the final factored form.
Here's an example to illustrate the process:
Let's say we have the trinomial 2x^2 + 7x + 3.
1. Multiplying 'a' and 'c', we get 2 * 3 = 6.
2. We need to find two numbers whose product is 6 and whose sum is 7. The numbers in this case are 3 and 2.
3. Now, we rewrite the middle term '7x' using these two numbers: 7x = 3x + 4x.
So, the trinomial becomes 2x^2 + 3x + 4x + 3.
4. Group the terms: (2x^2 + 3x) + (4x + 3).
5. Factor out the greatest common factor from each group: x(2x + 3) + 1(4x + 3).
6. Combine the factored groups: (2x + 3)(x + 1).
So, the factored form of 2x^2 + 7x + 3 is (2x + 3)(x + 1).
Remember that practice makes perfect when it comes to factoring trinomials. The more examples you work through, the better you'll become at recognizing patterns and finding the correct factors.
You must write your trinomial example:
2 x^2 − 5x + 3
x ^ 2 + 2x – 8
etc.