Question

Find each f(c) using synthetic substitution. f(x) = –x7 + 4x6 – 4x5 – 6x4 – 3x3 – 6x2 – x – 2; c = –5

Answer 1

Answer:

f(-5)=149603

Step-by-step explanation:

Synthetic Substitution

It's a method to find the value of a polynomial for a given value of x. The polynomial we want to evaluate is

$f(x) = -x^7 + 4x^6 - 4x^5 - 6x^4 - 3x^3 - 6x^2 - x - 2$

at x=-5

We can, of course, just replace x for -5 and compute the powers, products and sums and subtractions, but we can use the synthetic substitution to avoid the powers, just by multiplying and adding or subtracting. The method requires us to use only the coefficients of the ordered and complete polynomial as shown:

-1  4  -4  -6  -3  -6  -1  -2

Now we set up an arrangement where we can place some operations

     -1  4  -4  -6  -3  -6  -1  -2

-5

The second row and a new third row will be filled by computing values as explained below.

The first value of the third row is the very same first coefficient (-1)

     -1   4   -4   -6   -3   -6   -1   -2

-5

     -1

The second value of the second row is the -5 times the first value of the third row: (-5)(-1)=5

Next value of the third row is the sum of the values above of it: 4+5=9

-1 4 -4 -6 -3 -6 -1 -2

-5  5

-1 9

Now we proceed the same with next values to get the full arrangement or synthetic substitution

-1 4 -4 -6 -3    -6  -1  -2

-5  5 -45 245 -1195 5990 -29920 149605

-1 9 -49 239 -1198 5984 -29921 149603

The very last value of the third row is the required substitution

f(-5)=149603

The formatted table can be seen in the image below