Problem 19 - College Algebra

Question

Divide using synthetic division. $$ \left(3 x^{2}+7 x-20\right) \div(x+5) $$

Step-by-Step Solution

Answer

The solution to the synthetic division \((3 x^{2}+7 x-20) \div (x+5)\) is \(3x -8\) remainder \(20\).

Step 1: Set up the Synthetic Division

List the coefficients of the polynomial to be divided which are \(3, 7, -20\). In the front of these, write the number you get by making \(x+5=0\), which is \(-5\). Then draw a line underneath.

Step 2: Bring Down and Multiply

Bring down the first coefficient which is \(3\), under the line. Multiply this by \(-5\) (the number written outside) and write the result \(-15\) under the second coefficient \(7\), then add them together to get \(-8\) and write it under the line. This becomes the coefficient of the x term in the answer.

Step 3: Repeat the Process

Repeat the process: multiply the new number \(-8\) by \(-5\), write the result \(40\) under \(-20\), add them together to get \(20\), and write it under the line. This becomes the constant in the answer.

Step 4: Write the Answer

Write the answer using the numbers under the line and decrease the power of x by one. So, the answer is \(3x -8\) with a reminder of \(20\).

Key Concepts

Polynomial Division

Polynomial division is a method for dividing a polynomial by another polynomial, similar to the way we divide numbers. When dealing with polynomials, it's important to arrange them in standard form, which means writing terms in descending order based on their degree (the power of the variable).
There are several methods to perform polynomial division, such as long division and synthetic division. Synthetic division is a simplified method particularly useful when dividing a polynomial by a linear binomial of the form \( x - c \).
When applying synthetic division, follow these main steps:

• Identify the divisor (the binomial) and ensure it is in the form \( x - c \). For the problem \( x + 5 \), rewrite it as \( x - (-5) \) which means \( c = -5 \).
• List all coefficients of the dividend polynomial in order.
• Apply the synthetic division process to find the quotient and possible remainder.

This method offers a quick way to solve polynomial division problems while keeping calculations straightforward.

Coefficient Analysis

Coefficient analysis involves understanding and using the coefficients of a polynomial during division. In synthetic division, coefficients play a crucial role as they allow us to perform the division conveniently.
To carry out coefficient analysis in synthetic division:

• Extract the coefficients of the polynomial to be divided. For example, from \(3x^2 + 7x - 20\), the coefficients are \(3, 7, -20\).
• Use these coefficients systematically with the number \(c\) derived from the divisor (remember \( x + 5 \) gives \( c = -5 \)).
• Track each step: bring down the first coefficient, multiply it by \(c\) and add it to the next coefficient in line, continuing this process until complete.

Tracking coefficients through these operations is essential to verify each part of the process is done accurately, resulting in the correct quotient and remainder.

Remainder Theorem

The Remainder Theorem is a helpful mathematical concept that can be a powerful cross-verification tool while performing polynomial divisions. This theorem states that if a polynomial \( f(x) \) is divided by \( x - c \), then the remainder of this division will be \( f(c) \).
When using synthetic division, this theorem can confirm your results. Once the synthetic division process is complete, the final value below the line after the operations is the remainder. For example, if the result is \( 3x - 8 \) with a remainder of \(20\), this can be double-checked by plugging \(c\) back into the original polynomial. If \(f(-5) = 20\), you can be confident in the accuracy of your division.
The Remainder Theorem thus ensures not only a quick division method but also provides a way to confirm that the quotient and remainder derived are correct, adding another layer of comprehension and reliability to polynomial division.