Problem 49 - A Graphical Approach to Precalculus with Limits

Question

Use synthetic substitution to determine whether the given number is a zero of the polynomial. $$-0.5 ; \quad P(x)=4 x^{3}+12 x^{2}+7 x+1$$

Step-by-Step Solution

Answer

Yes, -0.5 is a zero of the polynomial.

Step 1: Set Up Synthetic Substitution

To use synthetic substitution, we first need to set up our synthetic division table. The coefficients of the polynomial \( P(x) = 4x^3 + 12x^2 + 7x + 1 \) are \( 4, 12, 7, 1 \). We will use \( -0.5 \) as the divisor.

Step 2: Perform Synthetic Division

Write the coefficients \( 4, 12, 7, 1 \) in a row. Bring down the first coefficient, \( 4 \), as is. Then multiply \( 4 \) by \(-0.5\) to get \(-2\), and add this to the next coefficient \( 12 \) to get \( 10 \). Repeat the process for all coefficients:1. Multiply \( 10 \) by \(-0.5\) to get \(-5\), add to \( 7 \) to get \( 2 \).2. Multiply \( 2 \) by \(-0.5\) to get \(-1\), add to \( 1 \) to get \( 0 \).The final row of numbers should be \( 4, 10, 2, 0 \).

Step 3: Determine the Remainder

The last number \( 0 \) from the synthetic division represents the remainder of the polynomial when divided by \( x + 0.5 \) (since \(-0.5\) would create a factor of \( x + 0.5 \)). If the remainder is \( 0 \), then \(-0.5\) is a zero of the polynomial.

Step 4: Conclude Whether the Number is a Zero

Since the remainder is \( 0 \), it means that \(-0.5\) is indeed a zero of the polynomial \( P(x) = 4x^3 + 12x^2 + 7x + 1 \).

Key Concepts

Polynomials

Polynomials are expressions comprised of variables and coefficients, combined using addition, subtraction, multiplication, and exponentiation with non-negative integer exponents. They are fundamental in algebra and appear in many forms such as quadratics, cubics, quartics, and more. To illustrate, a simple polynomial is given as:

• A linear polynomial: \( P(x) = ax + b \)
• A quadratic polynomial: \( P(x) = ax^2 + bx + c \)
• A cubic polynomial: \( P(x) = ax^3 + bx^2 + cx + d \)

In the case of our problem, we are working with a cubic polynomial \( P(x) = 4x^3 + 12x^2 + 7x + 1 \). The coefficients here are \( 4, 12, 7, \) and \( 1 \). These coefficients are crucial as they play a key role in operations like synthetic substitution or division. Understanding the structure of different polynomials helps in recognizing patterns and applying the right techniques for solving related mathematical problems.

Zeros of Polynomials

Zeros or roots of a polynomial are the values of the variable which make the polynomial equal to zero. In simpler terms, these are the solutions to the polynomial equation \( P(x) = 0 \). Zeros can be real or complex numbers and are fundamental in graphing polynomials, as they tell us where the graph intersects the x-axis. To find if a particular number is a zero of a given polynomial, such as \(-0.5\) in our exercise, we substitute it into the polynomial and simplify. If the result is zero, the number is a zero of the polynomial.For example, suppose \( 3 \) is a zero of the polynomial \( P(x) = x^2 - 5x + 6 \). Substituting \( 3 \) will yield \( P(3) = 3^2 - 5 \, \cdot \, 3 + 6 = 0 \). Thus, 3 is indeed a zero. Finding zeros is a significant task because they provide information about the polynomial's factorization and graph behavior.

Synthetic Division

Synthetic Division is a simplified method of dividing a polynomial by a binomial of the form \( x - c \). It is particularly useful for finding zeros and quickly evaluating polynomials. The process reduces the complexity of traditional long division and makes calculations more straightforward.Here's a step-by-step breakdown of synthetic division as used in our example:

• Set up the synthetic division: Write down the coefficients of the polynomial. Also, write the constant \(-c\) which is derived from the divisor \(x - c\).
• Perform calculations: Start with the leading coefficient, bring it down unchanged. Multiply it by \(-c\) and add it to the next coefficient, continuing the process to the end.
• Analyze the result: The last number in the row represents the remainder. If the remainder is zero, the divisor is indeed a factor, making \( c \) a zero of the polynomial.

In our exercise, we established that \(-0.5\) is a zero because the remainder was zero in our synthetic division process. Once mastered, Synthetic Division is a powerful tool for students tackling polynomial-related problems. It's essential for not only solving but also understanding deeper properties of polynomials.