Problem 19 - Algebra for College Students

Question

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(-3 x^{3}+x^{2}+2 x+2\right) \div(x+1) $$

Step-by-Step Solution

Answer

Quotient: \(-3x^2 + 4x - 2\), Remainder: \(4\).

Step 1: Set up synthetic division

First, write down the coefficients of the polynomial \(-3x^3 + x^2 + 2x + 2\). These coefficients are \(-3, 1, 2,\) and \(2\). For the divisor \(x+1\), set \(x+1 = 0\) which gives \(x = -1\). This is the number we'll use for the synthetic division.

Step 2: Perform synthetic division

Start the synthetic division by bringing down \(-3\), the leading coefficient, to the bottom row. Multiply \(-1\) (from \(x=-1\)) by \(-3\), and write the result \(3\) in the next column under the second coefficient, which is \(1\). Add \(1 + 3 = 4\). Continue this process: multiply \(-1\) by \(4\) to get \(-4\), add it to the next coefficient \(2\) to get \(-2\). Lastly, multiply \(-1\) by \(-2\) to get \(2\), and add it to \(2\) to get \(4\).

Step 3: Interpret the results

The bottom row of numbers \(-3, 4, -2, 4\) represents the coefficients of the quotient and the remainder. The quotient is \(-3x^2 + 4x - 2\) and the remainder is \(4\). Thus, the result of the division is \(-3x^2 + 4x - 2\) with a remainder of \(4\).

Key Concepts

Polynomial Division

Polynomial division is a process to divide two polynomials, somewhat similar to how we divide numbers. When using long division with polynomials, the goal is to express a polynomial as a quotient and a remainder with respect to another polynomial. However, this can become cumbersome.

That's where synthetic division comes in handy. It's a simplified form of polynomial division, specifically used when dividing by a linear factor of the form \(x - c\). Compared to traditional long division, synthetic division is usually faster and involves less writing.

• It involves writing down only the coefficients of the dividend polynomial.
• It simplifies the multiplication and subtraction steps.

Using synthetic division for the exercise, we work with the coefficients of \(-3x^3 + x^2 + 2x + 2\) and the divisor \(x + 1\) becomes \(x - (-1)\). By substituting \(-1\) into the synthetic division process, we obtain a quicker route to find the quotient and remainder.

Quotient and Remainder Theorem

The Quotient and Remainder Theorem is a fundamental principle in algebra. It states that when a polynomial \(f(x)\) is divided by a non-zero polynomial \(d(x)\), there are two resulting polynomials: a quotient \(q(x)\) and a remainder \(r(x)\). The relationship can be expressed as:

\[ f(x) = d(x) \cdot q(x) + r(x)\]
Here, \(r(x)\) has a lower degree than \(d(x)\). In the original exercise, \(-3x^3 + x^2 + 2x + 2\) is divided by \(x+1\), resulting in a quotient of \(-3x^2 + 4x - 2\) and a remainder of \(4\).

This theorem helps us confirm that our division process, synthetic or long, is indeed correct. It also allows us to reconstruct the original polynomial using the quotient, remainder, and divisor, which is a powerful check for the validity of your solution.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. They play a crucial role in mathematics, especially when simplifying and solving equations. When working with algebraic expressions, you're often required to perform operations such as adding, subtracting, multiplying, and dividing, as in the case of polynomial division.

Understanding how to manipulate algebraic expressions is key to mastering various math problems. With the division of polynomials, we can simplify a complex expression into simpler, more manageable parts: a quotient and a remainder.

• The coefficients (like \(-3, 1, 2, 2\) from our exercise) are vital in this simplification process.
• Upon completion, we express the result in terms of a new algebraic expression, \(-3x^2 + 4x - 2\), with the leftover part, the remainder \(4\).

This ability to simplify using algebraic rules supports solving equations and understanding algebra deeper.