Question

Discuss how to use synthetic division to evaluate a polynomial function.

Answer 1

Synthetic division is a method used to divide a polynomial function by a binomial of the form (x - a), where a is a constant. It is helpful in evaluating polynomial functions to find their roots or zeros. To use synthetic division, follow the steps outlined in the answer.

Step-by-step explanation:

Synthetic division is a method used to divide a polynomial function by a binomial of the form (x - a), where a is a constant. It is helpful in evaluating polynomial functions to find their roots or zeros.

To use synthetic division, follow these steps:

1. Arrange the terms of the polynomial function in descending order of exponents.
2. Write the constant term of the binomial divisor (x - a) as the first row of the synthetic division table.
3. Bring down the leading coefficient (the coefficient of the term with the highest power of x) as the first entry in the second row of the table.
4. Multiply the constant term in the first row by the entry in the second row, and write the result below the next term in the first row.
5. Add the entry in the first row to the result in the second row, and write the sum below the next term in the first row.
6. Repeat steps 4 and 5 until all terms have been exhausted.
7. The numbers in the last row of the table are the coefficients of the quotient polynomial.

For example, if we want to evaluate the polynomial function f(x) = 4x^3 - 7x^2 + 6x - 9 at x = 2, we can use synthetic division:

-2 (|) 4 -7 6 -9

|___________

8 -2 8 -10

___________________

4 1 14 -19

The numbers in the last row, 4, 1, 14, -19, are the coefficients of the quotient polynomial. Therefore, when x = 2, the value of f(x) is given by the quotient polynomial, which is 4x^2 + x + 14.