Problem 41 - College Algebra with Modeling and Visualization

Question

Use synthetic division to divide the first polymomial by the second. $$3 x^{3}-11 x^{2}-20 x+3 \quad\quad\quad x-5$$

Step-by-Step Solution

Answer

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

Step 1: Set Up Synthetic Division

The divisor is \(x - 5\), so we use \(5\) as our divider. Write down the coefficients of the dividend: \(3, -11, -20, 3\). Make sure to line these coefficients in a row.

Step 2: Bring Down the Leading Coefficient

Bring down the first coefficient \(3\) as it is, and write it beneath the line.

Step 3: Multiply and Add

Multiply the leading number under the line by \(5\) and write the result under the next coefficient. Add it to the coefficient above it (\(-11 + 15 = 4\)) and write the result below, creating the next value under the line.

Step 4: Repeat Multiplication and Addition

Take the new number \(4\) you just wrote, multiply by \(5\) (\(4 \times 5 = 20\)) and add this result to the next coefficient \(-20\). This gives you \(-20 + 20 = 0\). Write this under the line.

Step 5: Final Multiplication and Addition

Multiply the new result \(0\) by \(5\) (\(0 \times 5 = 0\)) and add this to the last coefficient \(3\). This gives \(3 + 0 = 3\). Write this beneath the line.

Step 6: Write the Result

The numbers under the line are \(3, 4, 0, 3\). These represent the coefficients of the quotient polynomial: \(3x^2 + 4x + 0\), with a remainder of \(3\). Therefore, the result of the division is \(3x^2 + 4x + 3/(x-5)\).

Key Concepts

Polynomial Division

When working with polynomial division, one of the methods we commonly use is synthetic division. This technique simplifies the division of a polynomial by a linear binomial of the form \(x - c\), where \(c\) is a constant. It's particularly useful because it streamlines calculations.

• Instead of lengthy long division, synthetic division enables more succinct calculations.
• Tackle problems step by step by focusing only on coefficients, not variables.
• Great for situations where the divisor is a simple binomial, like our divisor \(x - 5\).

In our problem, we took the cubic polynomial \(3x^3 - 11x^2 - 20x + 3\) and divided it by \(x - 5\). By focusing on the coefficients \(3, -11, -20,\) and \(3\), synthetic division provides an efficient process to arrive at the quotient polynomial.

Remainder Theorem

The remainder theorem is a crucial aspect of polynomial division, especially when using synthetic division. It states that if you divide a polynomial \(f(x)\) by a linear factor \(x - c\), the remainder of this division process is simply \(f(c)\).

• Helps easily verify division results.
• Connects directly with how synthetic division calculates remainders.
• Applies the given constant from the divisor to simplify finding remainders.

In our example dividing by \(x - 5\), we predicted the remainder would match the evaluation \(3(5)^3 - 11(5)^2 - 20(5) + 3\), which aligns perfectly with the remainder found through synthetic division, namely \(3\). This theorem supports our result, validating the division process and yielding quick confirmations.

Algebraic Expressions

Understanding algebraic expressions is fundamental when manipulating polynomials through division techniques. Each polynomial expression consists of terms that can include constants, variables raised to powers, and coefficients.

• Key elements like variables and coefficients are vital for operations.
• Handling algebraic expressions correctly is crucial for successful division.
• Identifies structure in expressions, allowing for strategic approaches in simplifying them.

In the exercise, the given polynomial \(3x^3 - 11x^2 - 20x + 3\) was structured to identify coefficients and powers of \(x\) for synthetic division. Recognizing and reorganizing terms as coefficients allowed us to conduct the division efficiently, highlighting the role of thoroughly understanding algebraic expressions and their components for accurate mathematical procedures.