Question

Use Horner's method to evaulate P(−3) and P'(−3), where P(x)=x⁴+4x²−3x+2.

Answer 1

To evaluate P(-3) and P'(-3) for the polynomial P(x), Horner's method organizes the computation in a tabular form and involves synthetic division. Write down the coefficients for P(x) and P'(x), bring down the leading coefficient, multiply by -3, add to the next coefficient, and repeat until the end.

Step-by-step explanation:

To evaluate P(-3) and P'(-3) for the polynomial P(x) = x⁴ + 4x² - 3x + 2 using Horner's method, we first perform the evaluation for P(-3). Horner's method reduces the computational effort by organizing the calculation in a tabular form and performing synthetic division.

For P(-3), proceed as follows:

1. Write the coefficients: 1, 0, 4, -3, 2 (Note that we include 0 for the x³ term which is not present).
2. Bring down the leading coefficient (1 in this case).
3. Multiply this coefficient by -3 and write the result below the second coefficient.
4. Add this result to the second coefficient and repeat this process until the last coefficient.

To find P'(-3), first find the derivative of P(x): P'(x) = 4x³ + 8x - 3. Then apply Horner's method similarly to how we evaluated P(-3).

Step-by-step for P'(-3):

1. Write the coefficients of P'(x): 4, 0, 8, -3.
2. Repeat the Horner's method process using these coefficients and -3 as the value.

The values obtained from the final line in both processes will give you P(-3) and P'(-3), respectively.