Use synthetic division and the Remainder Theorem to evaluate P(c).
P(x) = 5x5 + 10x3 + x + 1, c = −2
p(-2)=
You can see the steps here:
http://www.emathhelp.net/calculators/algebra-1/synthetic-division-calculator/?numer=5x^5+%2B+10x^3+%2B+x+%2B+1&denom=x%2B2&steps=on
But you will need to practice some of these on your own.
Well, synthetic division is a method used to divide a polynomial by a binomial of the form (x - c), where c is a constant. In this case, we want to use synthetic division to divide P(x) by (x - (-2)) or simply (x + 2).
Let's begin by setting up the synthetic division:
-2 | 5 0 10 1 1
--------------------
Now, bring down the first coefficient, which is 5:
-2 | 5 0 10 1 1
--------------------
5
Now, multiply -2 by 5 and write the result under the next coefficient, which is 0:
-2 | 5 0 10 1 1
--------------------
5
-10
Add the numbers in the second column:
-2 | 5 0 10 1 1
--------------------
5
-10
-----
-5
Multiply -2 by -5 and write the result under the next coefficient, which is 10:
-2 | 5 0 10 1 1
--------------------
5
-10
-----
-5
10
Add the numbers in the third column:
-2 | 5 0 10 1 1
--------------------
5
-10
-----
-5
10
-----
0
We have now successfully performed synthetic division. The remainders are written in the bottom row. In this case, since we obtained a remainder of 0, it means that (-2) is a root of the polynomial P(x).
Therefore, P(-2) = 0.
Funny how things work out sometimes, huh? So, the answer is 0.
To evaluate P(c) using synthetic division, we need to divide the polynomial P(x) by (x - c). In this case, we're evaluating P(-2), so we need to divide P(x) by (x + 2).
Using synthetic division, we set up the division as follows:
-2 | 5 0 10 1 0 1
|_________
We start by bringing down the 5:
-2 | 5 0 10 1 0 1
---------
5
Next, we multiply -2 by the 5 we just brought down, and write the result under the next coefficient:
-2 | 5 0 10 1 0 1
-10
---------
5
Then, we add the result to the next coefficient:
-2 | 5 0 10 1 0 1
-10
---------
0 0
We continue this process until we've gone through all the coefficients. The last number in our resulting row is the remainder.
In this case, the remainder is 0.
Therefore, P(-2) = 0.
To evaluate P(c) using synthetic division and the Remainder Theorem, follow these steps:
Step 1: Set up the synthetic division table
Arrange the coefficients of P(x) in descending order of the exponent of x. In this case, P(x) = 5x^5 + 10x^3 + x + 1, so the coefficients are: 5, 0, 10, 0, 1, 1.
| 5 0 10 0 1 1
-2 |
Step 2: Perform synthetic division
In the leftmost column of the table, write the value of c, which in this case is -2. Then, bring down the first coefficient (5) below the division bar.
| 5 0 10 0 1 1
-2 |
5
Multiply c (-2) by 5 and write the result (referred to as the partial product) above the next coefficient (0). Then, add the partial product to the coefficient and write the sum below the division bar.
| 5 0 10 0 1 1
-2 | -10
5
-10
Repeat the process with the next coefficients.
| 5 0 10 0 1 1
-2 | -10 20
5
-10 20
Next,
| 5 0 10 0 1 1
-2 | -10 20 -40
5
-10 20 -40
Continuing,
| 5 0 10 0 1 1
-2 | -10 20 -40 80
5
-10 20 -40 80
Finally,
| 5 0 10 0 1 1
-2 | -10 20 -40 80 -159
5
-10 20 -40 80 -159
Step 3: Interpret the result
The last number in the bottom row (-159) is the remainder. According to the Remainder Theorem, this remainder is equal to P(c), where c = -2. Hence, P(-2) = -159.