Problem 34 - Intermediate Algebra

Question

Use synthetic division to perform each division. See Example 3. Divide \(1-4 x+7 x^{2}+3 x^{3}\) by \(x+3\)

Step-by-Step Solution

Answer

The quotient is \(3x^2 - 2x + 2\) with a remainder of \(-5\). The expression is \(3x^2 - 2x + 2 - \frac{5}{x+3}\).

Step 1: Set up synthetic division

Synthetic division is used to divide polynomials in the form \(f(x) \div (x - c)\). Here, we are dividing by \(x + 3\), which can be rewritten as \(x - (-3)\). Therefore, the divisor is \(-3\). List the coefficients of the polynomial in descending order of powers: [3, 7, -4, 1].

Step 2: Perform division process

Write \(-3\) (the 'root' of the divisor) to the left. Draw a vertical bar and write the coefficients 3, 7, -4, 1 to the right. Bring down the first coefficient (3). Multiply this number by \(-3\) and write the result beneath the second coefficient. Add this result to the second coefficient (7) to get the new value.

Step 3: Continue synthetic division steps

Continue the process by multiplying the result from each column addition by \(-3\) and then adding this product to the next coefficient. Follow this procedure for each column: \(3 \times -3 = -9\), add to 7 to get \(-2\), continue with \(-2 \times -3 = 6\), add to \(-4\) to get 2, finally, \(2 \times -3 = -6\), add to 1 to get \(-5\).

Step 4: Interpret results

The numbers at the bottom represent the coefficients of the quotient polynomial, while the last number is the remainder. The result is therefore \(3x^2 - 2x + 2\) with a remainder of \(-5\). This means the division can be written as \(3x^2 - 2x + 2 - \frac{5}{x+3}\).

Key Concepts

Polynomial Division

Polynomial division is a method used for simplifying complex polynomial expressions by dividing them into simpler parts. It's much like long division with numbers, but here we deal with expressions that involve variables raised to powers, known as polynomials.

The main aim of polynomial division is to find how many times one polynomial (called the divisor) fits into another polynomial (called the dividend). The outcome consists of two parts: the quotient, representing how many times the divisor fits, and the remainder, what’s left over if it doesn't fit perfectly.

• The dividend is the polynomial you're dividing.
• The divisor is the polynomial you're dividing by.
• The quotient is the result of the division.
• The remainder is what's left after division.

This method is essential for understanding more advanced algebraic concepts and ties directly into synthetic division, making it more efficient when the divisor is a binomial.

Coefficients

Coefficients are the numerical part of the terms in polynomial expressions, providing them shape and form. They tell us the "size" or "scale" of the term they are a part of.

For example, in the term \( 7x^2 \), the number 7 is the coefficient. It's crucial to track these as they guide the values during polynomial operations like addition, subtraction, multiplication, and especially division.

• Arrange them in descending order of powers for clarity.
• They can be positive or negative numbers.
• During synthetic division, you place the coefficients in order and apply operations to them.

Note that while the coefficients are placed for organized synthetic division, their manipulation follows the rules of basic arithmetic. Knowing the coefficients and their order can help you predict the results and keep your arithmetic on track as you progress through a division problem.

Remainder Theorem

The Remainder Theorem offers a handy shortcut in polynomial division, stating that the remainder of the division of a polynomial \( f(x) \) by a linear divisor \( x - c \) is simply \( f(c) \).

What does this mean? If you substitute \( c \) in the polynomial \( f(x) \), the result you get will be the same as the remainder of the actual division. It works wonders for checking the accuracy of your calculations without performing complete division every single time.

• Great for double-checking your division results.
• Helps in identifying zeros or roots of polynomials.
• Useful in figuring out if a divisor divides the polynomial perfectly (remainder is zero).

When using synthetic division, this theorem allows you an easy check by verifying if the remainder corresponds to what \( f(c) \) would yield.

Division Algorithm in Algebra

The division algorithm in algebra is like a roadmap for dividing polynomials in a structured way. It’s essentially a formalized rule that guides us through polynomial division to achieve a quotient and remainder. According to the division algorithm, for any polynomials \( f(x) \) and \( d(x) \) (where \( d(x) \) is not zero), there exist unique polynomials \( q(x) \) (quotient) and \( r(x) \) (remainder), such that: \[ f(x) = d(x)q(x) + r(x) \]

Here, \( r(x) \) is always of lesser degree than \( d(x) \). This concept forms the backbone of understanding more advanced topics in algebra, such as factoring and solving polynomial equations.

• It ensures every polynomial division has a unique quotient and remainder.
• The remainder is smaller in degree than the divisor.
• Provides clarity and precision in polynomial arithmetic.

In essence, this algorithm is an algebraic translation of the long division process used in basic arithmetic but with powerful applications in higher math learning.