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Chapter 4: Problem 32

Using synthetic division, determine whether the numbers are zeros of the polynomial function. $$-4,2 ; f(x)=3 x^{3}+11 x^{2}-2 x+8$$

Short Answer

Expert verified
-4 is a zero; 2 is not a zero.

Step by step solution

01

- Setup Synthetic Division for \(-4\)

Write the coefficients of the polynomial function: \(3, 11, -2, 8\). Place \(-4\) on the left side and draw a synthetic division setup.
02

- Perform Synthetic Division with \(-4\)

Bring down the first coefficient, 3. Multiply \(3\) by \(-4\) and add to the next coefficient; repeat this process for each coefficient.
03

- Interpret Synthetic Division Result for \(-4\)

If the remainder (the final value at the bottom) is 0, then \(-4\) is a zero of the polynomial. If not, then \(-4\) is not a zero.
04

- Setup Synthetic Division for 2

Repeat the synthetic division setup using the coefficients \(3, 11, -2, 8\) and place 2 on the left side.
05

- Perform Synthetic Division with 2

Bring down the first coefficient, 3. Multiply \(3\) by 2 and add to the next coefficient; repeat this process for each coefficient.
06

- Interpret Synthetic Division Result for 2

If the remainder (the final value at the bottom) is 0, then 2 is a zero of the polynomial. If not, then 2 is not a zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

polynomial function
A polynomial function is a mathematical expression that involves a sum of powers of one or more variables multiplied by coefficients.
For example, in the polynomial function given in our exercise:
\( f(x) = 3x^{3} + 11x^{2} - 2x + 8 \),
  1. The highest degree (power of x) is 3, so it is a cubic polynomial.
  2. Each term consists of a coefficient and a variable raised to a power.
Polynomial functions can be represented in different forms, and they are commonly used to model a variety of real-world phenomena, such as economics, physics, and engineering. Understanding polynomials is essential for understanding complex mathematical concepts.
zeros of polynomial
Zeros of a polynomial, also known as roots or solutions, are values of the variable that make the polynomial equal to zero.
To find the zeros of a polynomial function \( f(x) \), we need to determine the values of x such that \( f(x) = 0 \). For the given polynomial: \( f(x) = 3x^{3} + 11x^{2} - 2x + 8 \), we need to check if \( -4 \) and \( 2 \) are zeros.
To check this, we can use synthetic division, which provides a quick method to determine if a given value is a zero of a polynomial.
synthetic division steps
Synthetic division is a simplified form of polynomial division. It is a method used to determine whether a given value is a zero of a polynomial function.
Here are the steps involved:
  1. Setup: Write down the coefficients of the polynomial (3, 11, -2, 8) and place the given value (-4 or 2) on the left side.
  2. Perform Division: Bring down the first coefficient. Multiply it by the given value and add the result to the next coefficient. Repeat this process for each coefficient.
  3. Interpret Result: If the remainder (the final value at the bottom) is 0, the given value is a zero of the polynomial. If not, it is not a zero.
For example, when checking if \(-4\) is a zero of \( f(x) = 3x^{3} + 11x^{2} - 2x + 8 \), we setup and perform synthetic division. If the final remainder is 0, then \(-4\) is a zero; otherwise it is not. Repeat the same process for the value \(2\).
Synthetic division is a powerful tool for quickly verifying potential zeros of polynomial functions, especially when dealing with higher-degree polynomials.

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