Hi I know is is probably basic but can anyone assist with the following :
Find all real Roots, expressing as whole # and # under radical sign, not as fraction. and factor completely
f(x)= X^3-2x^2-15x+30
I got roots at 2 (x-2) and +/- √5
If someone solves and gets factored as
(x-2) (x^2-15) can you show me line by line how to factor x^2-15?
I noticed that it factors quite easily using grouping
f(x) = x^2(x - 2) - 15(x - 2)
= (x-2)(x^2 - 15)
so if (x-2)(x^2 - 15) = 0
then x-2 = 0 or x^2 - 15 = 0
x = 2
or
x^2 - 15 = 0
x^2 = 15
x = ± √15
x^2 - 15 does not factor using rational numbers
you could look it as a difference of squares, and
(x-√15)(x+√15)
which would produce our other two roots of ± √15
picture of our situation
http://www.wolframalpha.com/input/?i=plot+y+%3D+x%5E3-2x%5E2-15x%2B30
notice it crosses the x - axis at 2 and around -3.9 and + 3.9
√15 = appr 3.9
Of course! I'll be happy to walk you through the process of factoring the expression x^2 - 15.
To factor x^2 - 15, we need to find two binomials whose product is equal to x^2 - 15. Let's proceed step by step:
Step 1: Determine the factors of 15
Find the pairs of numbers that multiply to give 15. The factors of 15 are 1 and 15, as well as -1 and -15.
Step 2: Look for the factors that also give -15 when their sum is taken
Since the coefficient of x^2 is positive (1 in this case) and the constant term is negative (-15), we are looking for two factors that have a difference of positive 15. After checking all the possible combinations, we find that the factors 5 and -3 satisfy this condition.
Step 3: Rewrite the middle term of the original expression
Replace the middle term (-15x) with the sum of the two factors we found in the previous step, which are 5 and -3. So, the expression becomes:
x^2 + 5x - 3x - 15
Step 4: Group the terms and factor by grouping
Now, we'll group the terms in pairs and look for their common factors. Group the first two terms (x^2 + 5x) and the last two terms (-3x - 15) separately:
(x^2 + 5x) + (-3x - 15)
Step 5: Factor out the greatest common factor from each group
From the first group, we can factor out x, and from the second group, we can factor out -3:
x(x + 5) - 3(x +5)
Step 6: Notice the common binomial factor
The terms (x + 5) appear in both groups, which means we can factor it out. So, the expression becomes:
(x - 3)(x + 5)
Now you have successfully factored the expression x^2 - 15 as (x - 3)(x + 5).
If you multiply this expression back together, you will see that it equals x^2 - 15. I hope this helps! Let me know if you have any further questions.