Problem 9 - College Algebra

Question

Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$ \frac{2 x^{3}+7 x^{2}+9 x-20}{x+3} $$

Step-by-Step Solution

Answer

The quotient \(q(x)\) is \(2x^2 + x + 12\) and the remainder r(x) is -56

Step 1: Set up the polynomial long division

First, write the division in the format of a long division: \(x+3) \, \overline{| \, 2x^3 + 7x^2 + 9x - 20}\)

Step 2: Divide the leading term

Divide the leading term of the numerator by the leading term of the denominator. In our case, this will be \(\frac{2x^3}{x}\) which is equal to \(2x^2\). Write this above the division bar, aligned to the \(x^2\) term.

Step 3: Multiply and subtract

Multiply \(2x^2\) by the denominator, and subtract the result from the original polynomial inside the division. So, subtract \((x+3)(2x^2) = 2x^3 + 6x^2\) from the original polynomial \(2x^3 + 7x^2 + 9x - 20\). This gives a new polynomial \(x^2 + 9x - 20\).

Step 4: Repeat steps 2 and 3 until the degree of the remainder is less than the degree of the denominator.

Repeat steps 2 and 3. The leading term \(x^2\) is divided by the leading term \(x\) to give \(x\). This is multiplied by the denominator and subtracted from the previous remainder. This gives a new polynomial, \(12x - 20\). Then, the leading term \(12x\) is divided by the leading term \(x\) to give \(12\). This is multiplied by the denominator and subtracted from the previous remainder. The remainder is now \(-56\), which is degree 0, so we stop here.

Step 5: Write the answer

The final quotient (\(q(x)\)) is the polynomial above the division, and the remainder (\(r(x)\)) is the final polynomial inside the division. So, the quotient is \(2x^2 + x + 12\) and the remainder is \(-56\).

Key Concepts

Quotient and Remainder

In polynomial division, we divide one polynomial by another to obtain two results: the quotient and the remainder. The quotient represents the number of times the divisor fits into the dividend without exceeding it, similar to integer division. Meanwhile, the remainder is what is left over after division.

For the problem given, when you divide the polynomial \(2x^3 + 7x^2 + 9x - 20\) by \(x+3\), the quotient is \(2x^2 + x + 12\) and the remainder is \(-56\). This means that \(2x^2 + x + 12\) is the polynomial that you get after the division process, and \(-56\) is the leftover part that cannot be further divided by \(x+3\).

We can confirm the correctness of the division by using the formula: \[ \text{dividend} = \text{divisor} \times \text{quotient} + \text{remainder} \]Plugging in our values, it becomes: \[ 2x^3 + 7x^2 + 9x - 20 = (x+3)(2x^2 + x + 12) - 56 \]This equation should hold true, ensuring the division was performed correctly.

Dividing Polynomials

Dividing polynomials is akin to dividing numbers. Like long division with numbers, polynomial long division involves dividing terms sequentially to isolate out smaller parts that contribute to the quotient and remainder. The aim is to find what to multiply the divisor by so that when you subtract it from the dividend, it reduces the degree of the polynomial in the division.

Here’s a step-by-step approach when dividing polynomials:

• Identify the leading term of the dividend and the divisor.
• Divide the leading terms to get the first term of the quotient.
• Multiply the divisor by this term and subtract it from the dividend.
• Repeat these steps with the resulting new polynomial until what remains has a lower degree than the divisor.

Dividing polynomials requires practice to manage the divisions and subtractions accurately. However, once grasped, it's a valuable tool for working with complex algebraic expressions.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable present in the polynomial. It gives insight into the behavior and the form of the polynomial. When dividing polynomials, understanding the degree of each can provide a roadmap for how the division will unfold.

For a polynomial \(ax^n + bx^{n-1} + cx^{n-2} + \ldots\), the degree is \(n\) if \(a eq 0\).

In the division process, you often reduce the degree at each step by aligning terms systematically, starting from the highest degree of the dividend and moving downwards until you reach a polynomial of a lesser degree than the divisor. This systematic reduction is crucial as it aligns each step by step operation to work efficiently.

In the example provided, the division begins with a polynomial of degree 3 \((2x^3)\) and a divisor of degree 1 \((x)\). With each step, we methodically reduce the degree until the remainder has a degree less than the degree of the divisor, completing the division process.

Polynomial Division Steps

To successfully execute polynomial long division, it's paramount to proceed methodically through a series of systematic steps. These steps ensure that the polynomial is reduced properly, and the results (the quotient and the remainder) are accurate.

Here is a condensed version of the steps:

• Write the division in long division format.
• Divide the leading term of the dividend by the leading term of the divisor.
• Place the quotient in its position above the division bar.
• Multiply the entire divisor by this quotient term and subtract the result from the dividend.
• Bring down the next term if necessary, forming a new polynomial.
• Repeat the process with the new polynomial formed until the remainder's degree is lower than the divisor's degree.

By following these organized steps, you can systematically break down even complex polynomial division problems into simpler tasks, handling them with confidence. Consistency in following these steps will improve accuracy and understanding of the division process.