(p - 1)(p - 2)(p - 3) all over p+3 multiply p² - 9 all over p² - 3p + 2
First, factor the numerator:
(p - 1)(p - 2)(p - 3) = p³ - 6p² + 11p - 6
Then, factor the denominator:
p² - 3p + 2 = (p - 1)(p - 2)
(p + 3) cancels out from both numerator and denominator, leaving:
(p³ - 6p² + 11p - 6) ÷ (p - 1)(p - 2)
Now we can use partial fraction decomposition to separate the fraction into simpler parts:
(p³ - 6p² + 11p - 6) ÷ (p - 1)(p - 2) = A ÷ (p - 1) + B ÷ (p - 2)
To solve for A and B, we can use common denominator:
A(p - 2) + B(p - 1) = p³ - 6p² + 11p - 6
Setting p = 1, we get:
A(-1) + B(0) = -4
A = 4
Setting p = 2, we get:
A(0) + B(1) = 2
B = 2
Therefore, the original fraction can be rewritten as:
4 ÷ (p - 1) + 2 ÷ (p - 2)
Simplifying, we get:
4 ÷ (p - 1) + 2 ÷ (p - 2) = (4p - 6) ÷ (p² - 3p + 2)
Finally, we can substitute this expression back into the original problem:
(p - 1)(p - 2)(p - 3) ÷ (p + 3) × (p² - 9) ÷ (p² - 3p + 2)
= (p - 1)(p - 2)(p - 3) ÷ (p + 3) × [(p - 3)(p + 3) ÷ (p - 1)(p - 2)]
= (p - 1)(p - 2)(p - 3) ÷ (p - 1)(p - 2)
= p - 3
Therefore, the final answer is p - 3.
To simplify the expression ((p - 1)(p - 2)(p - 3))/(p + 3) * (p² - 9)/(p² - 3p + 2), we can follow these steps:
Step 1: Simplify the numerator (p - 1)(p - 2)(p - 3).
The numerator can be simplified by multiplying the terms within the parentheses:
(p - 1)(p - 2)(p - 3) = (p² - 3p + 2)(p - 3) = p³ - 6p² + 11p - 6.
Step 2: Simplify the denominator (p + 3).
Since the denominator is already in its simplest form, we leave it as (p + 3).
Now we have (p³ - 6p² + 11p - 6)/(p + 3) * (p² - 9)/(p² - 3p + 2).
Step 3: Factor both (p² - 9) and (p² - 3p + 2) in the numerator and denominator.
p² - 9 can be factored as (p + 3)(p - 3), and p² - 3p + 2 cannot be factored further.
Substituting these factorizations, our expression becomes:
(p³ - 6p² + 11p - 6)/(p + 3) * [(p + 3)(p - 3)]/(p² - 3p + 2).
Step 4: Cancel out common terms.
We can cancel out (p + 3)'s in the numerator and denominator, and we can also cancel out the p - 3 from the numerator and denominator.
After canceling, we are left with:
(p - 6)/(p² - 3p + 2).
So, the simplified expression is (p - 6)/(p² - 3p + 2).