Problem 27 - Intermediate Algebra

Question

Use synthetic division to perform each division. See Example 2. $$ \frac{4 t^{3}-t-18}{t-2} $$

Step-by-Step Solution

Answer

The quotient is \(4t^2 + 8t + 15\) with a remainder of 12.

Step 1: Set Up Synthetic Division

First, write the coefficients of the polynomial in descending order of power. Identify the divisor as "\(+2\)" from \(t - 2\). The coefficients of the polynomial \(4t^3 + 0t^2 - t - 18\) are \([4, 0, -1, -18]\).

Step 2: Perform the Division

Place \(+2\) on the left of a vertical bar and the coefficients \([4, 0, -1, -18]\) to the right. Drop the first coefficient 4 to the bottom row. Multiply \(+2\) by the current bottom value and place this result under the next coefficient, then add. Repeat this for the entire row.

Step 3: Multiply and Add

1. Bring down the 4.2. Multiply \(2\) by 4 (first coefficient) to get 8.3. Add 0 (the next coefficient) and 8 to get 8.4. Multiply \(2\) by 8 to get 16.5. Add -1 (the next coefficient) and 16 to get 15.6. Multiply \(2\) by 15 to get 30.7. Add -18 and 30 to get 12.

Step 4: Interpret the Result

The numbers at the bottom represent the coefficients and remainder of the resulting polynomial: \([4, 8, 15, 12]\). This represents \(4t^2 + 8t + 15\) with a remainder of 12. Therefore, \(\frac{4t^3 - t - 18}{t-2} = 4t^2 + 8t + 15 + \frac{12}{t-2}\).

Key Concepts

Polynomial Division

Polynomial division is very much like long division you learned in elementary school, but instead of numbers, we deal with polynomials. When dividing a polynomial by a simpler polynomial,

• such as a monomial like \( t - 2 \), it's useful to transform the division process into something more straightforward.
• This transformation is known as synthetic division, a special method used when dividing by linear terms.

The primary advantage of synthetic division over traditional polynomial division is its efficiency, mainly because it requires fewer written steps and operations.
In synthetic division,

• each step involves basic operations like addition and multiplication.
• This makes synthetic division handy and quick for solving problems dealing with polynomials.

Remainder Theorem

The remainder theorem is a useful tool in algebra, especially when dealing with polynomials. It states that if a polynomial \( f(t) \) is divided by \( t - a \), the remainder of the division is simply \( f(a) \).
This theorem provides a shortcut to find the remainder without performing the entire division.

• In the context of synthetic division, it allows us to quickly identify what the remainder is by just plugging in the root of the divisor (here, it's \(2\)).
• The remainder is then the result of this substitution, being the last value obtained in the synthetic division process.

In the given exercise, after completing the synthetic division, we observed a remainder of 12,

• which confirms that \( f(2) = 12 \), validating the remainder theorem effectively.

Rational Expressions

Rational expressions involve ratios of polynomials, much like how fractions are ratios of integers. When dealing with rational expressions, division plays a crucial role. In the expression \( \frac{4t^3 - t - 18}{t-2} \),

• the numerator is the polynomial we aim to divide, and the divisor—which acts as the denominator—is \( t - 2 \).

Synthetic division helps simplify the expression of the rational function by breaking it down into a simpler polynomial plus a remainder over the divisor.
The result from the synthetic division was

• \( 4t^2 + 8t + 15 \) and a remainder of 12.

Thus, the rational expression can be written as a polynomial plus the fraction \( \frac{12}{t-2} \).
Understanding these components enables us to simplify, evaluate, or even graph rational expressions,

• making tasks that deal with these expressions much more manageable and intuitive.