Problem 15 - Intermediate Algebra

Question

Use synthetic division to perform each division. See Example 1. $$ \left(5 x^{2}-27 x+10\right) \div(x-5) $$

Step-by-Step Solution

Answer

The quotient is \(5x - 2\) with a remainder of 0.

Step 1: Set Up Synthetic Division

Identify the coefficients of the polynomial dividend, which are 5, -27, and 10. Use the zero of the divisor, which is 5 (from \(x - 5 = 0\), solve for \(x\)). Set up the synthetic division by writing 5 (zero of divisor) on the left, and the coefficients 5, -27, and 10 to the right.

Step 2: Begin Synthetic Division

Bring down the leading coefficient 5 to the bottom row. Multiply it by 5 (the number on the left) and write the result (25) under the next coefficient (-27).

Step 3: Add Down the Column

Add -27 and 25 to get -2. Write this under the line, in the same column.

Step 4: Multiply and Add Again

Multiply -2 (the new bottom number) by 5 (the number on the left) to get -10. Write -10 under the last coefficient (10), and then add to get 0.

Step 5: Write the Result

The bottom row, after completing the division, gives us the quotient. The quotient is \(5x - 2\) and the remainder is 0. Therefore, the division is exact.

Key Concepts

Polynomial Division

Polynomial division is akin to long division but designed specifically for polynomials. This method is often employed to divide a polynomial by a divisor of the form \(x - c\). There are various methods for polynomial division, including the standard long division technique and a streamlined variant called synthetic division. Here, synthetic division is used because the divisor is linear (of the form \(x - c\)), which simplifies calculations compared to standard long division.

• The process starts by arranging the coefficients of the dividend polynomial in order.
• The divisor's root is then used in the synthetic division setup.
• Each step involves multiplying and adding, thus reducing the polynomial's degree step-by-step.

Synthetic division particularly shines with its straightforward approach, minimizing errors and making the polynomial division process less cumbersome. This method yields a quotient and a remainder, just like its long division counterpart.

Polynomials

Polynomials are algebraic expressions comprising variables and coefficients, which embody the form \(a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0\). Each term in a polynomial has:

• A coefficient, like 5 in \(5x^2\).
• A variable raised to an exponent, as seen in \(x^2\) which indicates the degree of the term.
• A constant term, here illustrated by 10, which holds an exponent of zero.

In algebraic operations, polynomials can be added, subtracted, multiplied, and divided. With division specifically, polynomials require a thoughtful approach to ensure the results reflect accurate quotient and remainder expressions. This clarity is crucial for solving polynomial equations effectively.

Algebra

Algebra is a foundational branch of mathematics that deals with symbols and the rules for manipulating these symbols. It is the language through which we formulate and solve equations. Algebra allows for the expression of mathematical relationships, making sense of symmetries, and solving for unknowns. In the context of polynomial division, specifically synthetic division, algebra's rules guide the step-by-step process.Consider what happens when dividing \((5x^2 - 27x + 10)\) by \((x - 5)\):

• We view the task as solving for the expression \(5x^2 - 27x + 10 = (x-5) \cdot \, ? \, + \, R\).
• Algebra's rules direct us in finding a quotient, represented by the lower degree polynomial.
• The remainder \(R\) checks whether the solution is exact or if adjustments are necessary.

Synthetic division exemplifies algebra at work by providing a streamlined pathway to answer specific division tasks, reiterating algebra's power and utility in simplifying complex problems.