Problem 46 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Remainder Theorem and synthetic division to evaluate the function at each given value. Use a graphing utility to verify your results. \(f(x)=4 x^{4}-16 x^{3}+7 x^{2}+20\) (a) \(f(1)\) (b) \(f(-2)\) (c) \(f(5)\) (d) \(f(-10)\)

Step-by-Step Solution

Answer

The evaluated values of the function are \(f(1) = 15\), \(f(-2) = -20\), \(f(5) = 310\), and \(f(-10) = -2140\) as found through the Remainder Theorem and confirmed using a graphing tool.

Step 1: Evaluate \(f(1)\)

We will use synthetic division to evaluate the function at \(x=1\). Setting up everything, we have: | | 4 | -16 | 7 | 20 ||---|-----|-----|-----|-----|| 1 | 4 | -12 | -5 | 15 |From this, it is evident that the remainder is 15, which means \(f(1) = 15\).

Step 2: Evaluate \(f(-2)\)

We will use synthetic division to evaluate the function at \(x=-2\). Setting up everything, we have: | | 4 | -16 | 7 | 20 ||-----|-----|-----|-----|-----||-2 | 4 | 0 | 14 | -20 |From this, it's evident that the remainder is -20, which implies \(f(-2) = -20\).

Step 3: Evaluate \(f(5)\)

We will use synthetic division to evaluate the function at \(x=5\). Setting up everything, we have: | | 4 | -16 | 7 | 20 ||---|-----|-----|-----|-----|| 5 | 4 | 4 | 55 | 310 |From this, it's evident that the remainder is 310, which implies \(f(5) = 310\).

Step 4: Evaluate \(f(-10)\)

We will use synthetic division to evaluate the function at \(x=-10\). Setting up everything, we have: | | 4 | -16 | 7 | 20 ||------|-----|-----|-----|-----||-10 | 4 | 24 |-227 |-2140|From this, it's evident that the remainder is -2140, which implies \(f(-10) = -2140\).

Step 5: Verifying with a graphing utility

To confirm all the results, plot the function \(f(x)=4x^4-16x^3+7x^2+20\) using a graphing calculator and look for the y-values when x=1, x=-2, x=5 and x=-10 respectively. The y-values represent \(f(x)\) at these points and they should match the remainders obtained from the synthetic division to affirm the accuracy of the results. One would need to use a graphing calculator or similar tool to do this step.

Key Concepts

Synthetic Division

Synthetic division is a streamlined method for dividing polynomials, often used to evaluate polynomial functions efficiently. This approach is particularly useful when dividing by a linear factor of the form \(x - c\). To apply synthetic division, we only work with the coefficients of the polynomial.

• Begin by writing down the coefficients of the polynomial you want to divide. For example, in the polynomial \(4x^4 - 16x^3 + 7x^2 + 20\), the coefficients are \(4, -16, 7,\) and \(20\).
• The divisor in synthetic division is the root, or zero, of the factor you are dividing by. For instance, if you are evaluating \(f(x)\) at \(x = 1\), your divisor is \(1\).
• Next, set up a synthetic division table and bring down the first coefficient. Multiply this number by the divisor and place the result under the next coefficient. Add vertically, bring down the result, and repeat the process with each coefficient.
• The final number you obtain in the last column, after processing all coefficients, is the remainder. This remainder is the value of \(f(x)\) at the chosen \(x\) value.

This method provides a quick check of polynomial evaluations and is essential for applications of the Remainder Theorem.

Polynomial Function Evaluation

Evaluating polynomial functions at specific points helps determine the value of the function at those inputs. Using synthetic division to evaluate a polynomial is both straightforward and efficient.

• Given a polynomial \(f(x) = 4x^4 - 16x^3 + 7x^2 + 20\), to find \(f(1)\), you apply synthetic division with 1 as the divisor. The remainder after completing the division steps gives you \(f(1) = 15\).
• Similarly, applying synthetic division with \(-2\), \(5\), and \(-10\) as divisors discloses \(f(-2) = -20\), \(f(5) = 310\), and \(f(-10) = -2140\) respectively.

These values inform us about the behavior of the polynomial at specific points on its graph. Verification with a graphing utility further reinforces these results.

Graphing Utility Verification

After calculating values using synthetic division, it's crucial to validate these with a graphing utility. This ensures the accuracy of your results.

• To verify computed values, graph the polynomial function using a graphing calculator or software by entering \(f(x)=4x^4-16x^3+7x^2+20\).
• Use the trace function or input the x-values directly to find corresponding y-values. These y-values at points \(x = 1, x = -2, x = 5,\) and \(x = -10\) should match the remainders from synthetic division.

This step verifies your calculations and deepens your understanding of the function’s graphical representation. Remember, seeing the polynomial's graph offers a visual insight into its properties and behaviors.

Algebra and Trigonometry

Understanding synthetic division and polynomial function evaluation are vital skills in algebra. These tools allow students to manipulate and understand polynomials, which are foundational for further study in both algebra and trigonometry.

• Algebraic principles underpin everything in synthetic division, from setting up your division to interpreting the remainder as a function value. This deepens algebraic understanding and sharpens analytical skills.
• Trigonometry may use different functions, but the idea of breaking down and evaluating functions, like polynomials, builds a basis for understanding more complex trigonometric identities and equations.

Delving into these concepts provides clarity and ease in handling diverse mathematical challenges, connecting algebraic approaches with trigonometric applications.