What factoring technique would be used to factor 2x^2+5x+3?
this is factoring of quadratic trinomials
because of the small numbers, it should be relatively easy to "guess" what the factors are
since the first term is 2x^2 the factors must start as
(2x + ?)(x + ?)
and since they end with a 3 the factors must end in 1 and 3
so it is either (2x+1)(x+3) or (2x+3(x+1)
it is easy to mentally check which gives us the 5x in the middle.
Trial and error. Whole number factors of 2 and 3 are limited, so it wont take long.
(2x+3)(x+1) Geepers, first try.
if you need a systematic way of factoring these, the method of "decomposition" seems to be a popular procedure taught these days.
Are you familiar with that?
Not really
I will illustrate with an example
6x^2 + 41x - 30
multiply the coefficients of the first and last terms.... 6(-30) = -180
now look for factors of -180 which add up to +41 ,(obviously one is + the other - )
after a few tries you should find -4 and +45
so now "decompose" the middle term of 41x into -4x + 45x
6x^2 + 41x - 30
= 6x^2 - 4x + 45x - 30
= 2x(3x-2) + 15(3x-2)
= (3x-2)(2x+15)
oh that's what you mean. I guess we call it something different. Thanks
To factor the expression 2x^2 + 5x + 3, we can use a factoring technique called "the product-sum method." Here's how we can do it:
Step 1: Multiply the coefficient of the quadratic term (2) with the constant term (3). In this case, 2 * 3 = 6.
Step 2: Find two numbers that multiply to give us the result from step 1 (6) and add up to the coefficient of the linear term (5). In this case, the numbers are 2 and 3 because 2 + 3 = 5 and 2 * 3 = 6.
Step 3: Rewrite the middle term (5x) using the two numbers found in step 2. We split the middle term using these numbers. So, instead of 5x, we write 2x + 3x.
Now, the expression becomes: 2x^2 + 2x + 3x + 3.
Step 4: Group the terms and factor by grouping. We factor out the greatest common factor from each group.
The expression can now be written as: (2x^2 + 2x) + (3x + 3).
Step 5: Factor out the greatest common factor from each set of grouped terms.
(2x^2 + 2x) = 2x(x + 1)
(3x + 3) = 3(x + 1)
The factored form becomes: 2x(x + 1) + 3(x + 1).
Step 6: Combine the factored terms.
The final factored form is: (2x + 3)(x + 1).
So, the factored form of 2x^2 + 5x + 3 is (2x + 3)(x + 1).