Problem 34 - Algebra and Trigonometry

Question

25-38 . Find the quotient and remainder using synthetic division. $$ \frac{x^{3}-9 x^{2}+27 x-27}{x-3} $$

Step-by-Step Solution

Answer

Quotient: \( x^2 - 6x + 9 \), Remainder: 0.

Step 1: Setup for Synthetic Division

Identify the coefficients of the polynomial. The polynomial is \( x^3 - 9x^2 + 27x - 27 \), so the coefficients are \([1, -9, 27, -27]\). The divisor is \( x - 3 \), so we use \( 3 \) as the divisor.

Step 2: Perform Synthetic Division - Bring Down the Leading Coefficient

Write the number \( 3 \) on the left of the synthetic division setup and the coefficients \([1, -9, 27, -27]\) at the top. Bring down the \( 1 \) from the top as the first number on the bottom row.

Step 3: Multiply and Add

Multiply the divisor \( 3 \) by the number just written below (\( 1 \)) and place the result (\( 3 \)) below the next coefficient \(-9\). Add \(-9\) and \( 3 \) to get \(-6\), and write it below.

Step 4: Continue the Multiplication and Addition

Multiply \( 3 \) by the most recently written number \(-6\) to get \(-18\). Place \(-18\) below the next coefficient, \( 27 \). Add \( 27 \) and \(-18\) to get \( 9 \) and write \( 9 \) below.

Step 5: Final Multiplication and Addition

Multiply \( 3 \) by \( 9 \) to get \( 27 \). Place \( 27 \) below the last coefficient, \(-27 \). Add \(-27\) and \( 27\) to get \( 0 \), which is the remainder.

Step 6: Interpret the Results

The bottom row gives the coefficients of the quotient \( x^2 - 6x + 9 \) and the remainder is \( 0 \). Therefore, the quotient is \( x^2 - 6x + 9 \) and the remainder is \( 0 \).

Key Concepts

Polynomial Division

Polynomial division is similar to the long division process we learn in elementary school. It's used to divide a polynomial by another polynomial. The result is usually a combination of a quotient and possibly a remainder.

When tackled with synthetic division, it's a method particularly handy when dividing by a linear binomial of the form \(x - c\). The process becomes less tedious compared to the traditional methods like long division.

• Identify the polynomial you need to divide and its divisor. For synthetic division, the divisor should be in the format \(x - c\).
• Setup requires listing down the coefficients from the polynomial under division.
• The solution is obtained through a series of multiplications and additions.

Using synthetic division helps simplify operations by making them more systematic and compact. It also sheds light on the behavior of polynomials at specific points.

Remainder Theorem

The remainder theorem provides insight into the connection between division and evaluation of polynomials. According to this theorem, when you divide a polynomial \(f(x)\) by a linear divisor of the form \(x - c\), the remainder you get can also be found by evaluating \(f(c)\).

This means if you perform synthetic division on a polynomial using \(x - c\) as the divisor, the resulting remainder is precisely \(f(c)\).

• The remainder theorem simplifies various algebraic expressions when evaluating them at specific points.
• If your remainder is zero, it indicates that \(x - c\) is a factor of the polynomial.
• In our original exercise, once the division is performed, the remainder is zero, which aligns with the theorem.

Understanding this theorem helps in both evaluating and factorizing polynomials efficiently.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. These expressions are foundational to understanding polynomial behavior.

A polynomial is a special type of algebraic expression and can have various types. For instance, a polynomial of one term is a monomial, two terms form a binomial like \(x - c\), and three terms make a trinomial like \(x^2 - 6x + 9\).

• Polynomials are constructed from algebraic expressions by involving only whole number powers of the variables.
• Operations such as addition, subtraction, and multiplication are frequently applied to these expressions.
• In the synthetic division process, the expression being divided was a third-degree polynomial.

These algebraic fundamentals are essential since they play a crucial role in various fields such as engineering, physics, and computer science. Handling them effectively prepares students for advanced mathematics where understanding the structure and properties of such expressions is key.