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Chapter 4: Problem 26

Use synthetic division to find the function values. Then check your work using a graphing calculator. \(f(x)=2 x^{4}+x^{2}-10 x+1 ;\) find \(f(-10), f(2)\) and \(f(3)\)

Short Answer

Expert verified
f(-10) = 20101, f(2) = 17, f(3) = 142

Step by step solution

01

Set Up Synthetic Division for f(-10)

Write the coefficients of the polynomial: 2, 0, 1, -10, 1. Use -10 as the divisor.
02

Perform Synthetic Division for f(-10)

Bring down the first coefficient (2). Multiply by -10 and add to next coefficient: 2, (-10 * 2) + 0 = -20, (-10 * -20) + 1 = 201, (-10 * 201) - 10 = -2010, (-10 * -2010) + 1 = 20101. So the last number 20101 is f(-10).
03

Set Up Synthetic Division for f(2)

Write the coefficients of the polynomial: 2, 0, 1, -10, 1. Use 2 as the divisor.
04

Perform Synthetic Division for f(2)

Bring down the first coefficient (2). Multiply by 2 and add to next coefficient: 2, (2 * 2) + 0 = 4, (2 * 4) + 1 = 9, (2 * 9) - 10 = 8, (2 * 8) + 1 = 17. So the last number 17 is f(2).
05

Set Up Synthetic Division for f(3)

Write the coefficients of the polynomial: 2, 0, 1, -10, 1. Use 3 as the divisor.
06

Perform Synthetic Division for f(3)

Bring down the first coefficient (2). Multiply by 3 and add to next coefficient: 2, (3 * 2) + 0 = 6, (3 * 6) + 1 = 19, (3 * 19) - 10 = 47, (3 * 47) + 1 = 142. So the last number 142 is f(3).
07

Verify using Graphing Calculator

Input the polynomial function into the graphing calculator and evaluate f(-10), f(2), and f(3) to confirm they equal 20101, 17, and 142, respectively.

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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 consisting of variables and coefficients, structured in terms of powers of the variable. For example, the polynomial function given in the exercise is \(f(x)=2x^{4}+x^{2}-10x+1\). Each term is a product of a coefficient and a power of \(x\). Polynomial functions are widely used in algebra to represent complex relationships between variables.

In our example:
  • \(2x^4\) is a term with the coefficient 2 and the power 4.
  • \(x^2\) has the coefficient 1 and the power 2.
  • \(-10x\) has the coefficient -10 and the power 1.
  • 1 is a constant term.
Polynomials can have different degrees, determined by the highest power of the variable. Here, the degree is 4, making it a quartic polynomial function.
Graphing Calculator
A graphing calculator is a powerful tool used to visualize and solve mathematical problems, including evaluating polynomial functions. These calculators can graph functions, solve equations, and perform various algebraic computations.

When solving our problem, a graphing calculator can be used to:
  • Input the polynomial function \(f(x)=2x^4+x^2-10x+1\).
  • Evaluate the function at specific points, such as \(f(-10), f(2),\) and \(f(3)\).
  • Verify the results obtained through synthetic division.
Using a graphing calculator can provide a visual representation of the polynomial function and ensure the correctness of hand calculations.
Function Evaluation
Function evaluation involves finding the value of a function at a given input. For our polynomial function \(f(x)=2x^4+x^2-10x+1\), we want to find \(f(-10), f(2),\) and \(f(3)\).

To perform function evaluation using synthetic division:
  • Set up synthetic division with the given divisor (the value of x).
  • Write down the coefficients of the polynomial.
  • Use synthetic division to compute the result step-by-step.
For example, to find \(f(2)\) using synthetic division:
  • Write coefficients: 2, 0, 1, -10, 1.
  • Perform synthetic division with the divisor 2.
  • Follow steps to obtain the final result (in this case, 17).
Synthetic division simplifies function evaluation by breaking down the polynomial into simpler calculations, allowing us to efficiently find the desired values.

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Most popular questions from this chapter

Use synthetic division and the remainder theorem to find the zeros. $$f(x)=x^{3}-12 x+16$$

What does Descartes' rule of signs tell you about the number of positive real zeros and the number of negative real zeros of the function? $$P(x)=-3 x^{5}-7 x^{3}-4 x-5$$

What does Descartes' rule of signs tell you about the number of positive real zeros and the number of negative real zeros of the function? $$H(t)=5 t^{12}-7 t^{4}+3 t^{2}+t+1$$

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Total Profit. \(\quad\) Flexl, Inc., determines that its total profit is given by the function $$ P(x)=-3 x^{2}+630 x-6000 $$ a) Flexl makes a profit for those nonnegative values of \(x\) for which \(P(x)>0 .\) Find the values of \(x\) for which Flexl makes a profit. b) Flexl loses money for those nonnegative values of \(x\) for which \(P(x)<0 .\) Find the values of \(x\) for which Flexl loses money.
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