Sure, synthetic division is a method used to divide a polynomial by a binomial. It's a simpler method than long division when you're dividing by a linear term, and it allows you to quickly find the quotient and the remainder.
In this question, we have the polynomial F(x) = 2x³ - 10x - 28, and we want to find F(3) using synthetic division.
Step 1:
Write down the coefficients of the polynomial. The coefficients for our polynomial are 2 (for the x³ term), 0 (there's no x² term, so we put a 0 - this is an important step and shouldn't be ignored), -10 (for the x term), and -28 (for the constant term).
Step 2:
The root we are using for synthetic division is 3.
Step 3:
Start the synthetic division process: Carry the first coefficient (2 in our case) straight down.
Step 4:
Then multiply the number you just wrote down (2) by the root (3), getting 6.
Step 5:
Now add this number to the next coefficient (0), resulting in 6.
Step 6:
Repeat steps 4 and 5. Multiply the number you just wrote down (6) by the root (3) and add the next coefficient (-10). This gives 6*3 - 10 = 8.
Step 7:
Repeat steps 4 and 5 again. Multiply the number you just wrote down (8) by the root (3) and add the next coefficient (-28). This gives 8*3 - 28 = -4.
At the end of this process, the numbers we have down are 2, 6, 8, and -4.
The final number, -4, obtained at the end of synthetic division process is the value of F(3). Thus, F(3) = -4.
Hence, we have successfully used synthetic division to find the value of the polynomial at a specific point, x = 3, which turned out to be F(3) = -4.
Therefore, F(X)=2x^(3)-10x-28 = -4