Problem 45 - 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. \(h(x)=x^{3}-5 x^{2}-7 x+4\) (a) \(h(3)\) (b) \(h(2)\) (c) \(h(-2)\) (d) \(h(-5)\)

Step-by-Step Solution

Answer

The values for \(h(3)\), \(h(2)\), \(h(-2)\), and \(h(-5)\) based on the Remainder Theorem and synthetic division algorithm are -28, -12, -30, and -336 respectively.

Step 1: Apply synthetic division for \(h(3)\)

Set up the synthetic division grid: Draw a line, place '3' (the number you are dividing by) on the left, and write the coefficients of the polynomial to the right of the line in decreasing order of power (1, -5, -7, 4). Now, perform synthetic division: bring down the first coefficient ('1'), multiply 3 by 1 and write the result under the next coefficient making it '-2', add '-5' and '-2' giving '-7', multiply 3 by '-7' giving '-21', write this under the next coefficient making it '-28'. Hence, the remainder is -28.

Step 2: Apply synthetic division for \(h(2)\)

Set up the synthetic division grid similarly with '2'. Perform synthetic division: bring down the first coefficient '1', multiply 2 by 1 and write the result under the next coefficient making it '-3', add '-5' and '-3' giving '-8', multiply 2 by '-8' giving '-16', add '-16' and '4' giving '-12'. The remainder is -12.

Step 3: Apply synthetic division for \(h(-2)\)

Set up the synthetic division grid with '-2'. Perform synthetic division: bring down the first coefficient '1', multiply -2 by 1 and write the result under the next coefficient making it '-7', add '-5' and '-7' giving '-12', multiply -2 by '-12' giving '24', add '24' and '-7' giving '17', multiply -2 by 17 to get '-34', add -34 and 4 to get -30. The remainder is -30.

Step 4: Apply synthetic division for \(h(-5)\)

Set up the synthetic division grid with '-5'. Perform synthetic division: bring down the first coefficient '1', multiply -5 by 1 and write the result under the next coefficient making it '-10', add '-5' and '-10' giving '-15', multiply -5 by -15 giving '75', add '75' and '-7' giving '68', multiply -5 by 68 to get -340, add -340 and 4 to get -336. The remainder is -336.

Step 5: Verify Results

Use a graphing utility to graph the cubic function and check that the y-values at x = 3, 2, -2, -5 are equivalent to the rest obtained in steps 1-4.

Key Concepts

synthetic division

Synthetic division is a shorthand method used to divide a polynomial by a linear divisor of the form \(x-c\). This method is particularly useful because it simplifies the division process, allowing you to find the quotient and remainder quickly. Here's how to perform synthetic division:

• Write down the coefficients of the polynomial. For \(h(x)=x^3-5x^2-7x+4\), they are 1, -5, -7, and 4.
• Place the root of the divisor, \(c\), to the left of these coefficients. For example, for \(h(3)\), use 3.
• Bring down the leading coefficient to the bottom row.
• Multiply this number by the \(c\) value and write the product under the next coefficient.
• Add down the column, write the result on the bottom row, and repeat the process until all coefficients have been used.

Once completed, the last number in the bottom row is the remainder. In this exercise, we find remainders of -28, -12, -30, and -336 for the values 3, 2, -2, and -5, respectively. Synthetic division provides a quick and effective way to handle polynomial division problems.

polynomial evaluation

Polynomial evaluation is the process of determining the value of a polynomial for a given value of \(x\). The Remainder Theorem states that if a polynomial \(f(x)\) is divided by \(x-c\), then the remainder of this division is \(f(c)\). This theorem makes synthetic division an efficient method to evaluate polynomials.
Here's how it applies:

• Set a value for \(x\), such as 3, 2, -2, or -5 in our problem.
• Use synthetic division with the chosen \(x\) to find the remainder.
• This remainder will equal the polynomial's value at that specific \(x\).

For example, using the Remainder Theorem and synthetic division, the polynomial \(h(x)=x^3-5x^2-7x+4\) evaluates to -28 at \(x=3\). This method eliminates the need for manual substitution into \(h(x)\), providing quick results for polynomial values.

graphing utility verification

Graphing utility verification helps ascertain the accuracy of evaluated results by visualizing a polynomial function. It can confirm the outcomes obtained through synthetic division and the Remainder Theorem in a graphical format.
To use this tool:

• Graph the polynomial function using a graphing calculator or software.
• Examine the y-values that correspond to specific x-values of interest, such as those in steps 1-4 of our exercise.
• Check whether the y-values on the graph match the remainders calculated via synthetic division (e.g., -28 for \(x=3\), -12 for \(x=2\), etc.).

By visually confirming these calculations, graphing utilities make it easier to understand the behavior of complex functions. This step ensures the process' accuracy and provides insights into how the function behaves graphically.