Problem 79 - Intermediate Algebra

Question

Use synthetic division to perform each division. $$ \frac{4 x^{4}+12 x^{3}-x^{2}-x+12}{x+3} $$

Step-by-Step Solution

Answer

The quotient is \( 4x^3 - x + 2 \) with a remainder of 6.

Step 1: Write the Dividend and Divisor

The given polynomial to divide is \( 4x^4 + 12x^3 - x^2 - x + 12 \), and the divisor is \( x + 3 \). In synthetic division, we use the root of the divisor with a different sign. Here, we use \( -3 \) for the divisor \( x + 3 \).

Step 2: Set Up the Synthetic Division Table

Write the coefficients of the dividend: \( 4, 12, -1, -1, 12 \). Place these coefficients in a row. To the left of this row, write \( -3 \), the zero of the divisor.

Step 3: Start the Synthetic Division

Bring down the first coefficient, \( 4 \), to the bottom row as it is. This is the leading coefficient for the result.

Step 4: Multiply and Add

Multiply the number just written on the bottom row (\( 4 \)) by \(-3\), giving \(-12\). Write this under the next coefficient. Add \(12 + (-12) = 0\). Repeat this process for the next three coefficients.

Step 5: Complete the Synthetic Division

Continue the multiply and add process: Bring down the \( 0 \), multiply \( 0 \) by \(-3\) to get \(0\), add to -1 to get -1; then multiply \(-1\) by \(-3\) to get 3, add to -1 to get 2; then multiply \(2\) by \(-3\) to get -6, add to 12 to get 6.

Step 6: Write the Result

The numbers at the bottom now form \( 4, 0, -1, 2, \) and \( 6 \). These represent the coefficients for the result starting from one degree less than the original polynomial: \( 4x^3 + 0x^2 - x + 2 \) with a remainder of 6.

Key Concepts

Polynomial Division

When we deal with polynomial division, we're essentially splitting a polynomial (the dividend) by another polynomial (the divisor). Sometimes referred to as "long division" for polynomials, this concept is similar to numerical division, except it involves algebraic expressions.
The technique is crucial because:

• It helps simplify complex polynomial expressions.
• It allows us to rewrite polynomials in a different form, often revealing roots or factors.

In our exercise, we've used synthetic division, a shortcut method specifically for dividing polynomials of a form like \( x - k \). Unlike traditional polynomial long division, synthetic division simplifies calculations and requires less space. It's particularly handy when the divisors are simple linear polynomials, helping students better understand division without getting lost in lengthy arithmetic.

Remainder Theorem

The Remainder Theorem is a neat trick that relates to our division. It states that when a polynomial \( f(x) \) is divided by a linear divisor \( x - k \), the remainder of this division is \( f(k) \). This theorem is very efficient for determining the remainder quickly.
In our example:

• The divisor was \( x + 3 \), which we rewrite as \( x - (-3) \).
• The remainder found through synthetic division was 6.
• By applying the Remainder Theorem, evaluating the polynomial at \(-3\), we would directly calculate the remainder as well, without conducting full division.

This reflection helps cement the synthetic division result, showing another approach and verifying the consistency of polynomial properties.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations, forming the backbone of algebra studies. Understanding algebraic expressions is essential:

• They are fundamental in forming equations and functions.
• They allow us to represent real-world problems in a simplified way.

In polynomial division, knowing how to handle algebraic expressions like powers and coefficients is crucial. In our case, each term in \( 4x^4 + 12x^3 - x^2 - x + 12 \) has a coefficient associated with a respective power of \( x \).
Handling these correctly is important for the steps in synthetic division or any polynomial manipulation, allowing us to transform and simplify complex expressions into more manageable forms.