Problem 10 - Algebra for College Students

Question

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(2 x^{3}-7 x^{2}+2 x+3\right) \div(x-3) $$

Step-by-Step Solution

Answer

The quotient is \(2x^2 - x - 1\) with a remainder of \(0\).

Step 1: Set Up Synthetic Division

To use synthetic division, write the coefficients of the polynomial \(2x^3 - 7x^2 + 2x + 3\) in a row: \([2, -7, 2, 3]\). Use \(x-3=0\) to solve for \(x\), so the value used is \(3\).

Step 2: Bring Down the First Coefficient

Bring down the first coefficient \(2\) to the position directly below the line. This will be the first term of our result.

Step 3: Multiply and Add

Multiply the number you just brought down (\(2\)) by the divisor \(3\), giving \(6\). Add this product to the next coefficient \(-7\) to get \(-1\).

Step 4: Repeat Multiplication and Addition

Multiply \(-1\) by \(3\) to get \(-3\), then add it to the next coefficient \(2\). This results in \(-1\).

Step 5: Continue Multiplying and Adding

Multiply \(-1\) by \(3\), yielding \(-3\), and add it to the last coefficient \(3\). This results in \(0\).

Step 6: Interpret the Results

The final row of numbers \([2, -1, -1, 0]\) represents the coefficients of the quotient polynomial and the remainder. The quotient is \(2x^2 - x - 1\) and the remainder is \(0\).

Key Concepts

Polynomials

Polynomials are expressions composed of variables and coefficients, structured in terms of powers. In our example, the polynomial is given as:

• \( 2x^3 - 7x^2 + 2x + 3 \)

A polynomial of degree 3 in this case since the highest power of \( x \) is 3. Each term represents part of the polynomial and is constructed as the product of a constant (coefficient) and a variable raised to a power. Here, the coefficients are 2, -7, 2, and 3 respectively for each descending power of \( x \). Polynomials are important because they can represent a wide variety of functions and scenarios in algebra and are used extensively to model different real-world situations. Additionally, understanding polynomials is vital for delving deeper into algebraic functions, calculus, and beyond.
Breaking down terms in a polynomial helps in various mathematical operations including addition, subtraction, and particularly division.

Quotient and Remainder

The quotient and remainder are outcomes of the division process, whether performed with numbers or polynomials. In our context, using synthetic division to divide the polynomial \( 2x^3 - 7x^2 + 2x + 3 \) by \( x - 3 \), the quotient and remainder were successfully found. The quotient here is a polynomial of lower degree, noted as:

• \( 2x^2 - x - 1 \)

The remainder, which we seek at the end of our synthetic division process, was calculated to be 0. This means that \( x - 3 \) perfectly divides the polynomial without any left over. In polynomial division, understanding the quotient helps in systematizing polynomial expressions and simplifying complex polynomial equations. It indicates how many times the divisor can "fit" into the dividend entirely, while the remainder tells how much is "left over" after the division.

Algebra Steps

Synthetic division is a streamlined method primarily used for dividing polynomials, particularly when dividing by a linear factor. Let's review the algebra steps crucial to this method, which simplifies the division process significantly:

• Step 1: Set up the coefficients. Here, we took the coefficients \([2, -7, 2, 3]\) of the original polynomial.
• Step 2: Identify the divisor value. From \( x - 3 \), we determined \( 3 \) is used for multiplication.
• Step 3: Perform the division process. Begin by bringing down the initial coefficient (2), multiply, and then add sequentially to each subsequent coefficient.
• Step 4: Continue repeated multiplication and addition for each term, until you reach the final term.

The result of these steps gives us a transformed line of numbers, representing the new coefficients of the quotient and any remainder. The beauty of synthetic division lies in its simplification of polynomial division algebra steps, making it faster and cleaner than long division. This method is especially handy with large polynomials divided by simple binomials of the form \( x-c \).