Problem 44 - 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. \(g(x)=2 x^{6}+3 x^{4}-x^{2}+3\) (a) \(g(2)\) (b) \(g(1)\) (c) \(g(3)\) (d) \(g(-1)\)

Step-by-Step Solution

Answer

(a) \(g(2) = 175\), (b) \(g(1) = 7\), (c) \(g(3) = 1695\), (d) \(g(-1) = 7\)

Step 1: Evaluate g(2)

Substitute the given value (in this case 2) into the polynomial function \(g(x)\) to get the value of \(g(2)\). \[g(2)=2(2)^{6}+3(2)^{4}-(2)^{2}+3=2(64)+3(16)-4+3=128+48-4+3=175\]

Step 2: Evaluate g(1)

Substitute the given value (1) into the polynomial to get the value of \(g(1)\).\[g(1) = 2(1)^{6}+3(1)^{4}-(1)^{2}+3=2(1)+3(1)-1+3=2+3-1+3=7\]

Step 3: Evaluate g(3)

Substitute 3 for \(x\) in the polynomial \(g(x)\) to get \(g(3)\).\[g(3) = 2(3)^{6} + 3(3)^{4} - (3)^{2} + 3 = 2(729) + 3(81) - 9 + 3 = 1458 + 243 - 9 + 3 = 1695\]

Step 4: Evaluate g(-1)

Substitute -1 for \(x\) in the polynomial to find \(g(-1)\).\[g(-1) = 2(-1)^{6} + 3(-1)^{4} - (-1)^{2} + 3 = 2(1) + 3(1) - 1 + 3 = 2 + 3 - 1 + 3 = 7\]

Key Concepts

Synthetic Division

Synthetic division is a method used to divide polynomials. It is especially useful when you need to divide a polynomial by a linear divisor of the form \(x - c\). This technique streamlines the process and requires fewer computations compared to long division. It's particularly helpful in finding the remainder and quotient.Here are the simple steps for synthetic division:

• Identify the divisor, which in our exercise comes from each value like 2, 1, 3, and -1. This value is denoted as \(c\).
• Write down the coefficients of the polynomial in descending order of the power of \(x\).
• Draw a horizontal line and leave some space, and on the left side, write your value of \(c\). On the top, write the coefficients.
• Bring down the leading coefficient (the first number) without changing it.
• Multiply this leading coefficient by \(c\), place the result under the next coefficient, and add them together.
• Repeat the last step for all coefficients until you reach the end.
• The last number you calculate is the remainder.

Synthetic division is especially useful with larger polynomials. It is efficient and reduces potential errors during calculations.

Polynomial Evaluation

Polynomial evaluation involves calculating the value of a polynomial for a specific value of \(x\). This process is an essential step in many mathematical applications, including the Remainder Theorem.The process of polynomial evaluation can be straightforward:

• Take the polynomial, in this case, \(g(x) = 2x^6 + 3x^4 - x^2 + 3\).
• Substitute the given \(x\) value into the polynomial, replacing every \(x\) in the expression.
• Perform arithmetic operations, following the order of operations: exponents first, then multiplication, division, addition, and subtraction.

For instance, when evaluating \(g(2)\), you replace each \(x\) with 2, leading to calculations that result in 175 as demonstrated in the original steps. Polynomial evaluation helps in understanding the behavior of polynomials at different points and verifies solutions.

Graphing Utility

A graphing utility, such as a graphing calculator or software, is an advanced tool that visually represents functions and their behaviors.Graphing utilities serve several purposes:

• They can plot complex polynomials like \(g(x) = 2x^6 + 3x^4 - x^2 + 3\) across different values of \(x\).
• These tools allow for the visual examination of a polynomial's shape, helps identify roots, maxima, and minima, and can check for symmetry.
• Comparing the graph results with evaluated values (like the evaluations at \(x=2, 1, 3,-1\)) validates your manual calculations.

By using graphing utilities, you can gain additional insights into polynomials. Graphs offer a visual medium that complements algebraic computations, making complex relationships clearer and errors or inconsistencies easier to spot.