Problem 23 - Algebra for College Students

Question

Perform the indicated divisions. $$ \left(4 x^{3}-x^{2}-2 x+6\right) \div(x-2) $$

Step-by-Step Solution

Answer

Quotient: \(4x^2 + 7x + 12\); Remainder: 30.

Step 1: Set Up Long Division

Write the terms of the polynomial \(4x^3 - x^2 - 2x + 6\) under the division symbol, and \(x-2\) outside, as the divisor. Arrange the terms in descending order of their degrees.

Step 2: Divide the Leading Terms

Take the leading term of the dividend, \(4x^3\), and divide it by the leading term of the divisor, \(x\). This gives \(4x^2\). Write \(4x^2\) above the division line.

Step 3: Multiply and Subtract

Multiply \(4x^2\) by \(x-2\), giving \(4x^3 - 8x^2\). Write this beneath the dividend and subtract it. The new polynomial is \(7x^2 - 2x + 6\).

Step 4: Repeat Division Process

Divide the new leading term \(7x^2\) by \(x\), resulting in \(7x\). Multiply \(7x\) by \(x-2\) to get \(7x^2 - 14x\). Subtract to find the remainder, \(12x + 6\).

Step 5: Perform Final Division

Divide \(12x\) by \(x\), which equals \(12\). Multiply \(12\) by \(x-2\) getting \(12x - 24\). Subtract to arrive at the remainder, \(30\).

Step 6: Express the Solution

The result of the division is the quotient plus the remainder over the divisor. So, \(4x^2 + 7x + 12 + \frac{30}{x-2}\).

Key Concepts

Long Division Method

The Long Division Method is a technique used to divide polynomials, sequence by sequence, similar to the division of numbers. It is very systematic and involves multiple stages of division and subtraction, eventually leading to a quotient and a remainder.
To begin, arrange the terms of the dividend (the polynomial you are dividing) in descending order based on their degrees. This makes tracking the division process easier. For our example, we have the terms as follows: \[4x^3 - x^2 - 2x + 6.\] Next, take note of the divisor, which is a binomial: \[x - 2.\] Prepare the division as you would with numbers, placing the polynomial under the division symbol and the divisor outside.

• Start by dividing the leading term of your dividend by the leading term of the divisor.
• Write down the results step by step, as this helps avoid errors.

By following these steps, the Long Division Method simplifies the complex polynomial expressions into more manageable parts.

Polynomial Quotient

The Polynomial Quotient is the result you obtain after dividing one polynomial by another, similar to how dividing numbers produces a quotient.
In our example, the division takes the form of multiple steps, each producing a part of the quotient:

• First, we divide the highest degree term of the dividend, \(4x^3,\) by the highest degree term of the divisor, \(x,\) yielding \(4x^2.\)
• We continue the division process, tackling each subsequent term in the polynomial from highest degree to lowest.
• Each subsequent division produces the next term of the quotient.

Finally, the polynomial quotient of this process is \(4x^2 + 7x + 12,\) achieved through systematic reduction of terms using the Long Division Method. This does not account for any leftover remainder.

Remainder Theorem

The Remainder Theorem offers insight into polynomial division by stating that the remainder of dividing a polynomial \(f(x)\) by a linear divisor \(x - c\) is simply \(f(c).\) It provides a quick way to check your results from polynomial division and ensure accuracy.
In our exercise, after performing the Long Division Method, the remainder was found to be \(30.\) This was determined by subtracting the result of the multiplication in the final division step.

• The remainder is crucial because it completes the polynomial division process.
• It ensures you have accounted for all terms in your dividend equation.

Thus, applying the Remainder Theorem to the polynomial division gives the complete result: \(4x^2 + 7x + 12 + \frac{30}{x-2}.\)

Degree of Polynomial

The Degree of a Polynomial is the highest power of the variable x in the polynomial expression. It dictates the number of solutions the equation could have, as well as its general shape and characteristics.
For our dividend polynomial \(4x^3 - x^2 - 2x + 6,\) the highest degree is \(3,\) due to the term \(4x^3.\)

• The degree helps to determine the initial term of the quotient during division.
• It reflects how many reduction steps are needed as each subsequent degree is tackled until completion.

On the other hand, our divisor \(x - 2\) has a degree of \(1,\) typically simpler to manage during division. The understanding of degrees ensures one is aware of how to arrange the polynomial terms for efficient division.