Question

Explain how to determine the remainder when \(10 x^{4}-11 x^{3}-8 x^{2}+7 x+9\) is divided by \(2 x-3\) using synthetic division.

Short answer

The remainder is 15.

Step-by-step solution

- Set up the synthetic division

To use synthetic division, first express the divisor in the form of x - c. Here, the divisor is 2x - 3, so we have 2x - 3 = 0. Solving for x, we get x = 3/2. Our synthetic division will use 3/2 as the value to divide by. The coefficients of the polynomial are: 10, -11, -8, 7, and 9.

- Write down the coefficients

Write down the coefficients in a row: 10, -11, -8, 7, 9.

- Perform synthetic division

Start by bringing down the leading coefficient (10) to the bottom row. Then, multiply this number by 3/2 and write the result under the next coefficient. Add this result to the second coefficient and continue this process.

- Continue the synthetic division

After bringing down the leading coefficient (10), multiply 10 by 3/2 to get 15. Add 15 to -11 to get 4. Next, multiply 4 by 3/2 to get 6. Add 6 to -8 to get -2. Multiply -2 by 3/2 to get -3. Add -3 to 7 to get 4. Finally, multiply 4 by 3/2 to get 6. Add 6 to 9 to get 15.

- Write the remainder

The last number obtained in the synthetic division process is the remainder. Here, the remainder is 15.

Key concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another polynomial of a lower or equal degree. In this exercise, we focus on synthetic division, which is a simplified form of polynomial division when the divisor is a linear polynomial of the form x-c.

Synthetic division is a quicker and more efficient way to divide polynomials compared to the traditional long division method. It uses only the coefficients of the polynomials, making the process less cumbersome.

Let's break down how it is done:

• Identify the coefficient of each term of the dividend polynomial.
• Express the divisor in the form x-c and solve for x .
• Follow a systematic process of multiplying, adding, and bringing down coefficients.

In the given problem, we divided 10x^4 - 11x^3 - 8x^2 + 7x + 9 by 2x - 3 using synthetic division, and found the remainder to be 15. Synthetic division is particularly useful and efficient in finding such remainders.

Remainder Theorem

The remainder theorem is an important result related to polynomial division. It states that if a polynomial f(x) is divided by x-c, the remainder of this division is f(c).

This theorem helps in quickly evaluating polynomials without having to go through the entire division process. In our exercise, we find the remainder by performing synthetic division, which aligns with what the remainder theorem predicts.

Let's apply it to our specific problem:

• The polynomial is 10x^4 - 11x^3 - 8x^2 + 7x + 9.
• The divisor is 2x-3, which means x = 3/2.
• According to the remainder theorem, the remainder should be f(3/2).

By substituting 3/2 into the polynomial f(x) and evaluating it, you can verify that our synthetic division result matches this remainder. Hence, the remainder theorem provides a quick validation of the division process.

Precalculus

Precalculus is a course that prepares students for calculus. It encompasses a wide range of topics including functions, complex numbers, trigonometry, and, importantly for our exercise, polynomial and rational functions.

Understanding polynomial division and the remainder theorem is a key part of precalculus as it lays the groundwork for more advanced studies in calculus. Here are a few reasons why these concepts are vital:

• They develop algebraic manipulation skills.
• They are foundational for understanding limits and continuity in calculus.
• They help in solving polynomial equations and inequalities.

In our exercise, we applied these precalculus concepts to find the remainder when polynomial 10x^4 - 11x^3 - 8x^2 + 7x + 9 is divided by 2x - 3 using synthetic division. Mastery of these techniques is crucial for success in both precalculus and the subsequent study of calculus.