Problem 41 - Basic Technical Mathematics with Calculus

Question

Use synthetic division to determine whether or not the given numbers are zeros of the given functions. $$x^{4}-5 x^{3}-15 x^{2}+5 x+14 ; 7$$

Step-by-Step Solution

Answer

Yes, 7 is a zero of the polynomial.

Step 1: Setup Synthetic Division

First, list the coefficients of the polynomial. For the polynomial \(x^{4} - 5x^{3} - 15x^{2} + 5x + 14\), the coefficients are 1, -5, -15, 5, and 14. Write these down in order.

Step 2: Bring Down the First Coefficient

Begin synthetic division by bringing down the first coefficient, which is 1, to the bottom row.

Step 3: Multiply and Add

Now multiply the zero being tested (7) by the number you just brought down (1), getting 7, and add this to the next coefficient (-5), resulting in 2. Write 2 below the work line.

Step 4: Repeat Multiply and Add

Repeat the multiplication and addition process: Multiply 7 by the result from the previous column (2), getting 14. Add this to the next coefficient (-15), resulting in -1, and write it below the work line.

Step 5: Continue the Process

Continue this process. Multiply 7 by -1, getting -7, and add this result to the next coefficient (5), yielding -2. Write -2 below the line.

Step 6: Finish the Division

Finally, multiply 7 by -2, resulting in -14. Add this to the last coefficient (14), which gives a remainder of 0. Place a 0 below the final coefficient.

Step 7: Analyze the Remainder

Since the remainder is 0, this means the tested value, 7, is a zero of the polynomial. Synthetic division confirms that plugging in 7 results in a zero remainder.

Key Concepts

Zeros of a Function

To understand whether a number is a zero of a function, think of the zero as a value which, when substituted into the function, results in zero. For a polynomial function, such as \(x^4 - 5x^3 - 15x^2 + 5x + 14\), we want to find a number \(c\) for which the function \(f(c) = 0\). Finding zeros is crucial because they represent the points where the graph of the function intersects the x-axis.

• If you substitute the number into the polynomial and the result is 0, then that number is a zero of the function.
• Zeros are also known as roots or solutions, and they are important for graphing and understanding the function's behavior.

In the problem given, the number tested is 7. Using synthetic division here helps us verify this claim efficiently. If the process results in a remainder of zero, as it did here, it confirms that 7 is actually a zero of the function.

Polynomial Division

Polynomial division is similar to long division with numbers, but it involves dividing one polynomial by another. Synthetic division is a simplified form of polynomial division reserved for polynomials divided by linear polynomials of the form \(x - c\). It is especially appreciated for its speed and simplicity in testing potential zeros of polynomials.

• The process involves working through the coefficients of the polynomial, reducing the polynomial degree step-by-step.
• Synthetic division requires knowing the potential zero and arranging the polynomial's coefficients in order.

During the process of synthetic division:- We bring down the leading coefficient,- Multiply it by the supposed zero,- Add the results to the next coefficient, and- Continue the process until the end.In our exercise, synthetic division was used to confirm that 7 is a zero of the polynomial \(x^4 - 5x^3 - 15x^2 + 5x + 14\) by leaving a remainder of 0.

Remainder Theorem

The Remainder Theorem makes understanding polynomial division much simpler. It states that if a polynomial \(f(x)\) is divided by \(x - c\), the remainder is \(f(c)\). Thus, evaluating the polynomial at \(c\) gives the remainder of the division of \(f(x)\) by \(x - c\).This means when the remainder is zero, \(c\) is an actual zero of the polynomial. The theorem provides a fast way of checking whether a number is a zero without completely dividing the polynomial. In conjunction with synthetic division, it provides a very efficient method for testing potential zeros.In our given problem, the use of synthetic division and the outcome of a remainder of 0 confirms, according to the Remainder Theorem, that 7 is indeed a zero of the polynomial \(x^4 - 5x^3 - 15x^2 + 5x + 14\). This illustrates the elegance and power of the Remainder Theorem in action.