Problem 50 - A Graphical Approach to Precalculus with Limits

Question

Use synthetic substitution to determine whether the given number is a zero of the polynomial. $$-0.25 ; \quad P(x)=8 x^{3}+6 x^{2}-3 x-1$$

Step-by-Step Solution

Answer

Yes, -0.25 is a zero of the polynomial.

Step 1: Understand Synthetic Substitution

Synthetic substitution is a method used to evaluate a polynomial at a given value. If the polynomial evaluates to zero, the number is a zero of the polynomial.

Step 2: Set Up the Synthetic Substitution Table

First, set up the synthetic substitution table. Use the coefficients of the polynomial \(P(x) = 8x^3 + 6x^2 - 3x - 1\). The coefficients are \(8, 6, -3,\) and \(-1\). Set the value you are testing \(-0.25\) to the left. The table should look like this:\[\begin{array}{c|cccc}-0.25 & 8 & 6 & -3 & -1 \ & & & & \\end{array}\]

Step 3: Perform Synthetic Division Step 1

Bring down the leading coefficient, \(8\), to the bottom row. \[\begin{array}{c|cccc}-0.25 & 8 & 6 & -3 & -1 \ & & & & \\hline & 8 & & & \\end{array}\]

Step 4: Multiply and Add for the Next Column

Multiply \(-0.25\) by \(8\), giving \(-2\). Add \(-2\) to the next coefficient \(6\), resulting in \(4\).\[\begin{array}{c|cccc}-0.25 & 8 & 6 & -3 & -1 \ & & -2 & & \\hline & 8 & 4 & & \\end{array}\]

Step 5: Repeat Multiply and Add

Continue this process: Multiply \(-0.25\) by \(4\) to get \(-1\), add to \(-3\) to get \(-4\).\[\begin{array}{c|cccc}-0.25 & 8 & 6 & -3 & -1 \ & & -2 & -1 & \\hline & 8 & 4 & -4 & \\end{array}\]

Step 6: Final Step of Synthetic Division

Finally, Multiply \(-0.25\) by \(-4\) to get \(1\), add to \(-1\) to get \(0\). \[\begin{array}{c|cccc}-0.25 & 8 & 6 & -3 & -1 \ & & -2 & -1 & 1 \\hline & 8 & 4 & -4 & 0 \\end{array}\]

Step 7: Conclusion about Zero of the Polynomial

Since the remainder (last entry at the bottom row) is \(0\), \(-0.25\) is a zero of the polynomial.

Key Concepts

Polynomial Zero

A polynomial zero is a value of the variable that makes the entire polynomial equal to zero. In simple terms, if you substitute this number into the polynomial in place of the variable, you will get zero. Determining a polynomial's zeros is fundamental in solving polynomial equations and can give insight into the roots or solutions that satisfy the equation.

Consider the polynomial presented in the Original Exercise:
\[ P(x) = 8x^3 + 6x^2 - 3x - 1 \].
When evaluating whether \(-0.25\) is a zero of this polynomial, the goal is to check if substituting \(-0.25\) for \(x\) results in zero. This involves using a process called synthetic substitution. When the remainder of this process is zero, it confirms that \(-0.25\) is indeed a polynomial zero.

• Identifying zeros assists in graphing polynomials.
• Zeros indicate where the graph will intersect the x-axis.
• Zeros are essential for factoring polynomials and solving equations.

Synthetic Division

Synthetic division is a streamlined process for dividing a polynomial by a binomial of the form \(x - c\). This method is especially useful for quickly evaluating whether a given number is a zero of the polynomial, which means checking if the division results in a remainder of zero.

To perform synthetic division, you:

• Write down the coefficients of the polynomial.
• Place the number you're testing on the left side.
• Bring down the leading coefficient as the first number in the bottom row.
• Multiply the number on the left by the first number in the row, and add this product to the next coefficient.
• Repeat this process for each coefficient.

The final result of this division is in the last column. If this number is zero, it indicates that the number tested on the left is indeed a zero of the polynomial.

In our case, the number \(-0.25\) was found to give a remainder of zero after performing synthetic division with the polynomial \[P(x) = 8x^3 + 6x^2 - 3x - 1\]. This confirms the number is a zero.

Evaluating Polynomials

Evaluating polynomials means finding the value of a polynomial for a specific variable value. This is done by substituting the given value into the polynomial and performing the calculations. This process gives insight into the polynomial's behavior at the specific point or number.

In synthetic substitution, a form of evaluating polynomials, you're not only assessing the polynomial's value at a certain number but also testing if a number is a zero of the polynomial. Thus, you're concerned with whether the polynomial evaluates to zero.

To evaluate a polynomial using synthetic substitution, such as\[ P(x) = 8x^3 + 6x^2 - 3x - 1 \],and a specific value like \(-0.25\), follow the steps of synthetic division. The final result of this operation will show whether the evaluation is zero—indicating you found a zero of the polynomial—or if it results in another number, meaning the substitution is a standard evaluation.

• Evaluating helps confirm the roots discovered by analytical methods.
• It aids in checking solutions derived from equations logically.