Problem 37 - Algebra and Trigonometry

Question

25-38 . Find the quotient and remainder using synthetic division. $$ \frac{x^{3}-27}{x-3} $$

Step-by-Step Solution

Answer

Quotient: \(x^2 + 3x + 9\), Remainder: 0.

Step 1: Set up the Synthetic Division

Identify the divisor and ensure the dividend is arranged in descending order of power. Here, the divisor is \(x - 3\), and the dividend is \(x^3 - 27\). Use the root of the divisor, \(3\), for synthetic division. List the coefficients of \(x^3 - 27\), which are \(1, 0, 0, -27\).

Step 2: Perform Synthetic Division

Write the root of the divisor (\(3\)) to the left. Bring down the leading coefficient (\(1\)) directly since there is no need to multiply it by anything at first. Multiply it by the root, \(3\), and write the result (\(3\)) under the next coefficient (\(0\)). Add the column (\(0 + 3 = 3\)), and then continue this multiplication and addition process through the coefficients: Multiply \(3\) by \(3\) to get \(9\); add \(0 + 9 = 9\). Multiply \(9\) by \(3\) to get \(27\); add \(-27 + 27 = 0\).

Step 3: Identify the Quotient and Remainder

The last number is the remainder, and the other numbers are the coefficients of the quotient polynomial. Therefore, the quotient is \(x^2 + 3x + 9\) and the remainder is \(0\).

Step 4: Write the Final Answer

The original division can be expressed as \(x^3 - 27 = (x - 3)(x^2 + 3x + 9) + 0\), confirming that \(x^3 - 27\) is perfectly divisible by \(x - 3\).

Key Concepts

Polynomial Division

Polynomial division is a technique used to divide a polynomial by another polynomial, usually of a lower degree, to produce a quotient and possibly a remainder. There are two primary methods for polynomial division: long division and synthetic division. Each has its own uses and advantages.

• Long division is similar to numerical division but applied to polynomials, handling coefficients of terms systematically.
• Synthetic division is a shortcut method specifically useful when dividing by linear divisors like \(x - c\), where \(c\) is a constant.

Synthetic division significantly simplifies the division process, reducing time and complexity, especially for high-degree polynomials. It only uses the coefficients of the polynomial and involves a simpler calculation than long division.
Understanding polynomial division is crucial because it lays the foundation to solve polynomial equations. It also aids in simplifying complex polynomial expressions and functions.

Quotient and Remainder

When dividing polynomials, just like with numerical division, we find a quotient and possibly a remainder. The quotient is the result of the division prior to considering the remainder.

• In the given problem, the quotient is found to be \(x^2 + 3x + 9\).
• The remainder is the leftover part after the division. Here, it's \(0\), meaning the initial polynomial is perfectly divisible by \(x - 3\).

The process of finding the quotient and remainder is simplified by using synthetic division, which involves easy multiplication and addition.
The result can be expressed clearly as: \[x^3 - 27 = (x - 3)(x^2 + 3x + 9) + 0\] This formula helps verify the correctness of the division process, and shows whether the polynomial fully divides without leaving any remainder.

Roots of Polynomials

Understanding the roots of a polynomial is crucial in polynomial division because roots are essentially the solutions to the polynomial equation equals zero. Polynomials can have multiple roots, and identifying them can help simplify the division process.

• The root of the divisor \(x - 3\) is \(3\), as the equation \(x - 3 = 0\) directly gives \(x = 3\).
• This root is then used in synthetic division to simplify the whole division process.

Finding the root initially determines which polynomial can divide the given polynomial evenly, helping you find out if the polynomial is a factor.
The roots also inform us of any intersections with the x-axis on the graph of the polynomial function, indicating where the function changes signs.