How do you factor 2x^2-5x+2?
(x-2)(2x-1)
how do you figure that out?
trinomials like that can usually be done intuitively, but you should have been taught a method that you can use for general cases.
Look at the first term of 2x^2 and think where that can come from?
(x...)(2x...) is the only possibility.
Now look at the last term of +2
your brackets must have ended with +1 and +2 or -1 and -2
but surely they had to be both negative since the middle term is negative.
so now your choices are
(x-1()2x-2) or (x-2)(2x-1)
mentally expand them, only one gives you the correct middle term of the original.
So there is no way to incorporate the middle term from the beginning, like when there is no number in front of the x^2? And the only way to do it is keep guessing and checking?
You're right, when there is no number in front of the x^2 term, factoring becomes a little trickier. In such cases, you would have to use a method called trial and error or guessing and checking to find the correct factorization. Here's a step-by-step explanation of how you can approach factoring trinomials with no coefficient in front of the x^2 term:
1. Write down the trinomial: 2x^2 - 5x + 2.
2. Identify all possible factor pairs of the constant term (in this case, 2). In this case, the pairs are (1, 2) and (-1, -2).
3. Now consider the product of the x^2 coefficient (2) and the constant term (2), which is 4. You need to find a factor pair of 4 that sums up to the coefficient of the x term (-5). In this case, the pair is (-4, -1).
4. Rewrite the middle term (-5x) as the sum of two terms, using the factor pair found in the previous step. In our example, this would be -4x - x.
5. Group the terms into two pairs: (2x^2 - 4x) + (-x + 2).
6. Factor out the greatest common factor from each pair. In this case, the factor would be 2x from the first pair, and -1 from the second pair: 2x( x - 2) - 1(x - 2).
7. Notice that both pairs have a common factor of (x - 2).
8. Write down the factored form of the expression using the common factor from step 7: (x - 2)(2x - 1).
So to answer your question, when there is no number in front of the x^2, the trial and error method is commonly used to factorize the trinomial by finding factor pairs and checking if they correctly sum up to the coefficient of the x term.