SOLUTION: hi i am a tenth grade student needing help in algebra 2. my directions say Use synthetic division and the remainder theorem to find P(a) the problem is {{{ P(x)= x^3+7x^2+4x}}

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: hi i am a tenth grade student needing help in algebra 2. my directions say Use synthetic division and the remainder theorem to find P(a) the problem is {{{ P(x)= x^3+7x^2+4x}}      Log On


   

Question 230380: hi i am a tenth grade student needing help in algebra 2. my directions say Use synthetic division and the remainder theorem to find P(a)

the problem is +P%28x%29=+x%5E3%2B7x%5E2%2B4x;a=-2
Found 2 solutions by user_dude2008, jsmallt9:
Answer by user_dude2008(1862) About Me  (Show Source):
You can put this solution on YOUR website!
-2  |   1   7   4  0
    |
    ---------------------




-2  |   1   7   4  0
    |
    ---------------------
        1




-2  |   1   7   4  0
    |      -2
    ---------------------
        1   5




-2  |   1   7   4   0
    |      -2  -10  12
    ---------------------
        1   5  -6  12


Answer: P(a)=12

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
+P%28x%29=+x%5E3%2B7x%5E2%2B4x for P(a) we will divide P(x) by x-a using synthetic division. For a = 2:
2 |   1   7   4   0
---       2  18  44       
     ---------------
      1   9  22  44

Since our remainder is 44, P(2) = 44.

To understand why this works, let's think about division in general.
Let P(x) = any polynomial function
Let (x-a) = the divisor
Let Q(x) = the quotient you get from dividing P(x) by (x-a)
Let R/(x-a) = the remainder you get from dividing P(x) by (x-a). (Note: With a divisor like (x-a) R will just be some number. There will be no x's in the remainder.)
In other words:
P%28x%29%2F%28x-a%29+=+Q%28x%29+%2B+R%2F%28x-a%29
Let's multiply both sides by (x-a):
%28x-a%29%28P%28x%29%2F%28x-a%29%29+=+%28x-a%29%28Q%28x%29+%2B+R%2F%28x-a%29%29
On the right side we need to use the Distributive Property:
%28x-a%29%28P%28x%29%2F%28x-a%29%29+=+%28x-a%29%2AQ%28x%29+%2B+%28x-a%29%28R%2F%28x-a%29%29
Now we can cancel out some (x-a)'s:

leaving:
P%28x%29+=+%28x-a%29%2AQ%28x%29+%2B+R
Therefore
P%28a%29+=+%28a-a%29%2AQ%28a%29+%2B+R
Since (a-a) is zero and zero times anything is zero and zero + R is R:
P%28a%29+=+R
This is why the remainder, after dividing a polynomial, P(x), by (x-a) is the value of P(a). And synthetic division is a quick, simple, compact way to do this type of division.