In Exercises 55 - 58, use the Remainder Theorem and...

In Exercises 55 - 58, use the Remainder Theorem and synthetic division to find each function value. Verify your answers using another method.

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

(a) $ f(1) $ (b) $ f(-2) $ (c) $ f\left(\frac{1}{2}\right) $ (d) $ f(2) $

Key Concepts

Polynomial Evaluation

Polynomial evaluation involves determining the value of a polynomial function for a given input value. This concept is central to understanding the behavior of polynomial functions and can be executed through direct substitution or by employing methods like the Remainder Theorem combined with synthetic division.

Synthetic Division

Synthetic division is a simplified form of polynomial division that is particularly useful when dividing by linear factors. It streamlines the calculation process by using a tabular method, reducing the work involved compared to long division, and is commonly applied when using the Remainder Theorem.

Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor (x - c), the remainder of the division is equal to f(c). This theorem provides a quick method to evaluate polynomials at specific points and is a fundamental concept in polynomial algebra.

Transcript

00:01 Okay, i am going to use synthetic division to find each of these function values because we know according to the remainder theorem that the remainder from synthetic division is equal to the function value.

00:12 So for part a, for the synthetic division, we put one outside the box, and then we have 2x cubed, 0x squared, negative 7x, and 3.

00:23 We bring down the 2, multiply it by 1, write it in the next space, and add the column.

00:29 Multiply that by 1, write it in the next space and add the column.

00:33 Multiply that by 1, write it in the next space and add the column.

00:37 So the remainder is negative 2.

00:39 So that means that f of 1 is negative 2.

00:45 For part b, we do the same thing to find f of negative 2.

00:51 Negative 2 goes outside the box.

00:54 2, 0, negative 7, and 3 go in a row.

00:57 Bring down the first number.

00:59 Multiply it by negative 2, write it down and add.

01:02 Multiply that by negative 2, write it down and add.