Problem 30 - College Algebra and Calculus: An Applied Approach

Question

Use synthetic division to divide. Divisor \(x+3\) Dividend $$5 x^{3}$$

Step-by-Step Solution

Answer

Therefore, the result of the division \(5x^{3}\) by \(x+3\) is \(5x^2 - 15x + 45\) with remainder -135.

Step 1: Set Up the Problem

We first rewrite the divisor \(x+3\) in the form \(x - (-3)\), because synthetic division is set up to work with subtraction. The dividend \(5x^3\) can be rewritten as \(5x^3 + 0x^2 + 0x + 0\), to represent coefficients for all potential terms of the dividend.

Step 2: Set Up the Synthetic Division

We create a synthetic division table with -3 (from the divisor \(x-(-3)\) ) to the left of the vertical bar. Then, we fill in the coefficients of the dividend on the right of the vertical bar, which are 5, 0, 0, 0 in our case.

Step 3: Perform the Synthetic Division

First, bring down the leading coefficient, 5, to the bottom row. Then multiply -3 and 5 to get -15, and place it under the next coefficient, 0. Add these numbers to get -15, repeat this process until you reach the end of the dividends coefficients. So we will have three -15 on the bottom row.

Step 4: Interpret the Result

The bottom row of numbers gives the coefficients of the quotient and the remainder. Here, it will be \(5x^2 - 15x + 45 - 135/x+3\) which simplifies to \(5x^2 - 15x + 45\) with remainder -135.

Key Concepts

Polynomial Division

Polynomial division is a fundamental concept in algebra that allows us to divide one polynomial by another, generally of a lower degree. This technique is useful because it simplifies complex expressions and helps find roots of polynomial equations. Traditionally, polynomial division is similar to long division, just with variables. The division can be split into the following steps:

• Identify the divisor and dividend: The dividend is what you are dividing, and the divisor is by what you are dividing.
• Align the polynomial terms to ensure all powers of the variable are represented in both dividend and divisor. This might mean including terms with zero coefficients.
• Subtract corresponding terms to make a new polynomial with a smaller degree, repeatedly until the degree of the remainder is less than the degree of the divisor.
• Combine all intermediate quotients to form the final quotient polynomial.

When the traditional method gets cumbersome, synthetic division is a simpler alternative for dividing polynomials of the form (x - a). It involves using the roots of the divisor and operating mainly with coefficients. This makes it a quick and effective method.

Remainder Theorem

The Remainder Theorem is a significant principle that connects the remainder of a polynomial division with the value of the polynomial itself. It states that when a polynomial \(p(x)\) is divided by \(x - a\), the remainder of this division is \(p(a)\).Let's break down its usage:

• Consider the polynomial you want to divide, say \(p(x)\).
• Select the value \(x = a\) provided by the divisor \(x-a\).
• Instead of synthetically or traditionally finding the entire quotient, evaluate the polynomial at \(x = a\), \(p(a)\), which directly gives the remainder.

This theorem confirms that if a polynomial is completely divisible by \(x - a\), then the remainder is zero, implying \(x = a\) is a root of the polynomial. It's extremely useful when verifying roots or when division is unnecessary beyond finding the remainder.

Quotient and Remainder

When dividing polynomials, particularly through synthetic division, our goal is to find both the quotient and the remainder. Synthetic division helps in rapidly achieving this while avoiding the lengthy steps involved in long division.Here's how it works:

• After setting up the synthetic division, the numbers in the bottom row show coefficients for the quotient polynomial.
• The sequence of operations includes multiplying the divisor's root with the running total, then adding to the next coefficient.
• The final number on the bottom row is the remainder when the degree of the new polynomial is less than the divisor.

For example, a calculation might result in coefficients forming a polynomial like \(5x^2 - 15x + 45\) with a numerical remainder like -135. No terms remain leftover from division when the remainder's degree is lower than the divisor, confirming completion. This understanding allows for solving higher-degree problems efficiently through synthetic division.