Problem 30 - A Graphical Approach to College Algebra

Question

Find each quotient when \(P(x)\) is divided by the binomial following it. $$P(x)=-3 x^{3}-x-5 ; \quad x+1$$

Step-by-Step Solution

Answer

The quotient is \(-3x^2 + 3x - 4\) with a remainder of \(-1\).

Step 1: Set up the Division

To divide the polynomial \(P(x) = -3x^3 - x - 5\) by the binomial \(x + 1\), we use synthetic division. Since the divisor is \(x + 1\), the zero is \(-1\). We will use this in our synthetic division. The coefficients of \(P(x)\) are \(-3, 0, -1, -5\). The term \(0x^2\) is included as a placeholder.

Step 2: Perform Synthetic Division

Write \(-1\) on the left and the coefficients \(-3, 0, -1, -5\) on the right. Bring down the leading coefficient, \(-3\). Then, multiply \(-1\) by \(-3\) and write the result \(3\) underneath the next coefficient \(0\). Sum the column to get \(3\). Multiply \(-1\) by \(3\) to get \(-3\), write it under \(-1\), and sum to get \(-4\). Multiply \(-1\) by \(-4\) to get \(4\), put it under \(-5\), and sum to get \(-1\).

Step 3: Interpret the Result

The results from synthetic division give us the coefficients of the quotient and the remainder. The quotient from the division is \(-3x^2 + 3x - 4\), and the remainder is \(-1\). Thus, \(P(x) = (x+1)(-3x^2 + 3x - 4) - 1\).

Key Concepts

Synthetic Division

The technique of synthetic division is a simplified form of polynomial division. Unlike the longer traditional method, synthetic division provides a more straightforward way to divide polynomials, especially when dividing by a linear binomial like \(x + 1\). In this case, the zero of the binomial, which is \(-1\) (because \(x + 1 = 0\)), plays a crucial role in the process. By using the coefficients of the polynomial and systematically multiplying and adding them with the zero, we can quickly find the quotient and remainder.

Synthetic division is not only faster but also eliminates the need for writing variables and powers during the process. It greatly simplifies the task when dealing with complex polynomials.

Quotient

When dividing polynomials, one of the main results we look for is the quotient. The quotient is the polynomial that results from dividing one polynomial by another. In our example, after performing synthetic division on \(-3x^3 - x - 5\) with \(x + 1\), we find the quotient to be \(-3x^2 + 3x - 4\).

The quotient is expressed as a polynomial of one lower degree than the original polynomial \(P(x)\). This is due to the nature of division, where dividing a polynomial of degree \(n\) by a binomial results in a polynomial of degree \(n-1\). Understanding this concept helps us accurately decompose a polynomial into parts.

Remainder

The remainder in polynomial division is analogous to the remainder found when dividing numbers. It represents what is left over once the division is complete. In our polynomial division case, the remainder came out to be \(-1\).

Checking the remainder is crucial as it determines the completeness of the division process. If the remainder is zero, it means the divisor is a factor of the original polynomial. In this case, since the remainder is \(-1\), \(x + 1\) is not a factor of \(-3x^3 - x - 5\). Thus, the original polynomial can be expressed as:

\[P(x) = (x+1)(-3x^2 + 3x - 4) - 1\]

Understanding the remainder not only helps in verifying our work but also provides insight into the factorization of the polynomial.

Polynomials

Polynomials are expressions consisting of variables raised to various powers and multiplied by coefficients. Key elements include the terms, which are separated by addition or subtraction, where each consists of a product of a coefficient and a power of a variable.

In the exercise, our polynomial \(-3x^3 - x - 5\) is of degree 3, indicating that the highest power of \(x\) is 3. Polynomials can take various forms and complexities, yet each can be broken down into simpler elements through processes like division, which we applied here.

Understanding the structure of polynomials helps in simplifying them effectively, making it easier to manipulate them using various algebraic techniques such as addition, subtraction, multiplication, and particularly in this case, division. Being familiar with these concepts and operations is fundamental in algebra and calculus.