Problem 51 - A Graphical Approach to Precalculus with Limits

Question

Use synthetic substitution to determine whether the given number is a zero of the polynomial. $$-5 ; \quad P(x)=8 x^{3}+50 x^{2}+47 x+15$$

Step-by-Step Solution

Answer

Yes, -5 is a zero of the polynomial.

Step 1: Set Up Synthetic Substitution

First, write down the number you want to test ( -5) on the left-hand side. Then, list the coefficients of the polynomial on the right: 8, 50, 47, and 15.

Step 2: Perform Synthetic Substitution

Bring down the first coefficient (8) unchanged. Multiply -5 by 8, resulting in -40, and write it under the next coefficient, 50. Add 50 and -40, resulting in 10, and write it below. Continue this process of multiplying by -5, writing the result underneath the next coefficient, and adding, until you reach the last term.

Step 3: Calculate the Final Result

Continuing the steps from the previous part: Multiply -5 by 10 (resulting in -50), add to 47 (getting -3), multiply -5 by -3 (resulting in 15), and finally, add to 15 (getting 0).

Step 4: Determine if -5 is a Zero

The synthetic substitution process ends with a remainder of 0. Since the remainder is 0, -5 is indeed a zero of the polynomial.

Key Concepts

Polynomial

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Think of it as a line of numbers and letters strung together with plus and minus signs, reflecting a pattern based on a variable and its powers. For instance, in our exercise, the polynomial given is \( P(x) = 8x^3 + 50x^2 + 47x + 15 \).

• The highest power of the variable (in this case, \(x^3\)) determines the degree of the polynomial. Here, it's a cubic polynomial because the highest power of \(x\) is 3.
• Simplifying a polynomial means writing it in a format with the terms ordered by the degree of the variable, starting from the highest degree to the lowest.

Breaking down a polynomial into simpler parts often helps in solving it or understanding its properties. Polynomials serve as the building blocks for many algebraic operations and are foundational in mathematical models across diverse fields such as physics, engineering, and economics.

Coefficient

Coefficients are essential components of polynomials. They are the numerical factors that multiply the variables or their powers. In our polynomial, each term has a coefficient:

• In \(8x^3\), the coefficient is 8.
• In \(50x^2\), the coefficient is 50.
• In \(47x\), the coefficient is 47.
• In the constant term 15, the coefficient can be considered as 15.

Coefficients help determine the shape and direction of the polynomial's graph. They also have applications in synthetic substitution, where they are pivotal in easily determining whether a given value might be a zero of the polynomial.

Zero of a Polynomial

The zero of a polynomial is a value of the variable that makes the polynomial equal to zero. This means when you substitute this value into the polynomial, the result is zero. In the problem, we used synthetic substitution to check if -5 is a zero of the polynomial.

Understanding zeros is crucial because:

• Zeros represent the x-values where the polynomial graph intersects the x-axis.
• Finding zeros can help solve equations by reducing complex equations into simpler factors.
• A polynomial of degree \(n\) can have a maximum of \(n\) zeros.

In the exercise, we concluded that -5 is indeed a zero because, after performing synthetic substitution, the remainder was zero, confirming that \(P(-5) = 0\). Identifying zeros is a key skill for graph interpretation and complex problem-solving.