I missed a lesson today on factoring and was wondering if i can get some help here
for example, let's use 2x^2+5x+3
There are several ways.
The method that for some reason seems to be the most popular
these days is "decomposition"
In that method, multiply the coefficients of the x^2 term and the constant
.... (2)(3) = 6
Now look for factors of 6 that have a sum of +5
1,6 ---- 7 no
2,3 ---- 5 , got it!
now replace the 5x with 2x+3x
2x^2 + 2x + 3x + 3
factor in pairs:
2x(x+1) + 3(x+1)
now you have a common factor of x+1
= (x+1)(2x+3)
and there you have it!!!
Here is another example, a bit more complicated
8x^2 - 14x - 15
--- (8)(-15) = -120
factors:
-1,120 ---sum--- 119 , no
-2,60 ---sum--- 58
-4,30 ---sum--- 26
-5,24 ---sum--- 19
-6, 20 ---sum--- 14 , but 6,-20 has a sum of -14
8x^2 - 14x - 15
= 8x^2 - 20x + 6x - 15, if the signs are mixed I place the negative term first
= 4x(2x - 5) + 3(2x - 5)
= (2x - 5)(4x + 3)
Reiny, that was an excellent explanation!!
Of course! I can help you with factoring the quadratic expression 2x^2 + 5x + 3. Factoring is the process of breaking down an expression into simpler terms. In this case, we want to break down the quadratic expression into a product of two binomials.
To factor the quadratic expression, we need to find two binomials in the form (ax + b)(cx + d) that, when multiplied together, result in the original expression.
Here's how you can factor the expression step by step:
Step 1: Write down the expression in the standard form, which is ax^2 + bx + c. In this case, our expression is already in standard form: 2x^2 + 5x + 3.
Step 2: Look for the product of the leading coefficient (the coefficient of the x^2 term) and the constant term (the term without any variable) of the expression. In this case, the product is 2 * 3 = 6.
Step 3: We now need to find two numbers whose product is equal to the number we found in step 2 (6), and whose sum is equal to the coefficient of the middle term (5). In this case, the numbers are 2 and 3, because 2 * 3 = 6 and 2 + 3 = 5.
Step 4: Rewrite the middle term (5x) in the expression using the numbers we found in step 3. So, 5x can be written as 2x + 3x.
Now our expression becomes:
2x^2 + 2x + 3x + 3.
Step 5: Group the expression into two pairs, so that we have the common factors in each pair. In this case, we can group the expression as follows:
(2x^2 + 2x) + (3x + 3).
Step 6: Factor out the common factors from each pair. We can factor out 2x from the first pair, and 3 from the second pair. So, we are left with:
2x(x + 1) + 3(x + 1).
Step 7: Notice that both terms in the expression now have a common factor of (x + 1). Factor out the common binomial (x + 1) from the expression. Doing so, we get:
(x + 1)(2x + 3).
Therefore, the factored form of the quadratic expression 2x^2 + 5x + 3 is (x + 1)(2x + 3).
Remember to always double-check your answer by multiplying the two binomials together to make sure you get the original quadratic expression.