Problem 85 - Intermediate Algebra

Question

Use synthetic division to perform each division. $$ \left(4 x^{3}-1+5 x^{2}\right) \div(x+2) $$

Step-by-Step Solution

Answer

Quotient: \(4x^2 - 3x + 6\), Remainder: \(-13\).

Step 1: Arrange the Polynomial

Write the polynomial in standard form. The correct order of the terms for the polynomial is \(4x^3 + 5x^2 - 1\).

Step 2: Setup Synthetic Division

For synthetic division, we need to work with the root of the divisor. Since the divisor is \(x+2\), the root is \(-2\). Write \(-2\) on the left and the coefficients of the polynomial (\(4, 5, 0, -1\)) on the right.

Step 3: Bring Down the First Coefficient

Bring down the first coefficient, which is \(4\), to start the synthetic division.

Step 4: Perform Division Calculation

Multiply \(-2\) by \(4\) and write the result under the next coefficient (\(5\)). Add \(5 + (-8) = -3\). Repeat this process: Multiply \(-2\) by \(-3\) to get \(6\), add \(0 + 6 = 6\), multiply \(-2\) by \(6\) to get \(-12\), and finally add to \(-1\) to get \(-13\).

Step 5: Interpret the Result

The numbers obtained through synthetic division represent the coefficients of the quotient polynomial and the remainder. So, \(4x^2 - 3x + 6\) is the quotient, and \(-13\) is the remainder.

Key Concepts

Polynomial Division

Polynomial division is an essential algebraic process used to divide one polynomial by another, resulting in a quotient and occasionally a remainder. It's quite similar to long division but specifically tailored for polynomials. Synthetic division is a streamlined version used when dividing a polynomial by a divisor of the form \(x - c\). Here's how it works:

• First, ensure your polynomial is in standard form, with terms ordered by decreasing degree.
• Identify the root of the divisor. For \(x + 2\), the root is \(-2\).
• Use the coefficients of the polynomial to perform the division.

Synthetic division simplifies calculations and reduces chances of error compared to traditional polynomial division methods. Through these organized steps, it allows for the swift determination of the quotient and remainder, enhancing your proficiency in intermediate algebra.

Remainder Theorem

The Remainder Theorem is a handy tool in algebra that simplifies finding the remainder of a polynomial division. It states that when a polynomial \(f(x)\) is divided by \(x - c\), the remainder is \(f(c)\). This means that you can substitute the value of \(c\) into the polynomial to find the remainder quickly.While the synthetic division gives us a complete insight into the quotient and remainder like in our exercise, the Remainder Theorem provides a quick check for the remainder itself. For instance, in our example, substituting \(-2\) for \(x\) in \(4x^3 + 5x^2 - 1\) would directly yield \(-13\) as the remainder. Incorporating this theorem in your studies helps you verify division results efficiently without redoing extensive calculations, bolstering your confidence in polynomial operations.

Intermediate Algebra

Intermediate algebra serves as a bridge between basic algebraic principles and advanced topics, introducing students to more complex concepts such as synthetic division and the Remainder Theorem. This stage is crucial for developing a stronger algebraic foundation. In this phase of learning, students encounter:

• Polynomials and their characteristics.
• Division algorithms and techniques like synthetic division.
• The application of the Remainder Theorem to verify and solve problems.

Understanding intermediate algebra equips you with the skills to tackle higher-level mathematics, making it essential for academic progress. By mastering these topics, students can solve polynomial equations more effectively and feel more prepared for future challenges in mathematics.