Problem 7 - Algebra and Trigonometry

Question

\(3-8=\) Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express \(P\) in the form \(P(x)=D(x) \cdot Q(x)+R(x)\) $$P(x)=x^{4}-x^{3}+4 x+2, \quad D(x)=x^{2}+3$$

Step-by-Step Solution

Answer

Quotient: \(x^2 - x - 3\), Remainder: \(7x + 11\).

Step 1: Set Up the Long Division

Write down the dividend polynomial \(P(x) = x^4 - x^3 + 4x + 2\) and the divisor polynomial \(D(x) = x^2 + 3\). Prepare to perform polynomial long division by arranging terms in descending order of power.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \(x^4\) by the leading term of the divisor \(x^2\) to get the first term of the quotient: \(x^2\).

Step 3: Multiply and Subtract

Multiply \(x^2\) by \(D(x)\), which is \((x^2 + 3)\), resulting in \(x^4 + 3x^2\). Subtract this result from \(P(x)\) to find the new dividend: \(-x^3 - 3x^2 + 4x + 2\).

Step 4: Repeat the Process

Divide the new leading term \(-x^3\) by \(x^2\) to get \(-x\). Multiply \(-x\) by \(D(x)\) to get \(-x^3 - 3x\). Subtract this from the new dividend, resulting in \(-3x^2 + 7x + 2\).

Step 5: Continue Division

Divide \(-3x^2\) by \(x^2\) to get \(-3\). Multiply \(-3\) with \(D(x)\), obtaining \(-3x^2 - 9\). Subtract this from the existing dividend, yielding \(7x + 11\) as the remainder.

Step 6: Express the Final Result

The quotient from this division process is \(Q(x) = x^2 - x - 3\) and the remainder is \(R(x) = 7x + 11\). The polynomial \(P(x)\) can be expressed as \(P(x) = D(x) \cdot Q(x) + R(x) = (x^2 + 3)(x^2 - x - 3) + (7x + 11)\).

Key Concepts

Synthetic Division

Synthetic division is a streamlined method of dividing a polynomial by a binomial of the form \(x-a\). While not applicable for every polynomial division scenario, it's an efficient and simpler process for these specific cases. Instead of writing out all the terms of the dividend and divisor, synthetic division uses coefficients, reducing clutter.
Here's how it works:

• Identify the coefficients of the polynomial you wish to divide.
• The binomial's root \(a\) is used in place of the divisor.
• Set up a synthetic division array with these coefficients.
• Systematically perform the operations leading to the quotient and remainder.

This technique is particularly useful when you're looking for speed and efficiency, especially on problems involving large polynomials.

Long Division

Long division is a meticulous method for dividing any polynomial by another polynomial. This process extends the idea of numerical long division into the realm of algebra. While it might seem lengthy, it remains reliable and versatile for any type of polynomial division.
The steps include:

• Write the dividend and divisor in descending powers of the variable.
• Divide the first term of the dividend by the first term of the divisor, creating the first term of the quotient.
• Multiply and subtract to find the new dividend.
• Repeat until the degree of the new dividend is less than the divisor.

Through this method, students can always arrive at the complete expression of the quotient and remainder, ensuring accurate results.

Remainder Theorem

The Remainder Theorem offers a quick insight when dividing polynomials. It tells us that for a polynomial \(P(x)\) divided by \(x-a\), the remainder is simply \(P(a)\).
This theorem can save a lot of time in problem solving as:

• You only need to substitute \(a\) into \(P(x)\) to find the remainder.
• No division process is necessary to identify the rest of the polynomial expression.

It's a great tool to verify results from other division methods, ensuring correctness of your calculations easily.

Quotient

The quotient in polynomial division is akin to that in regular number division. It represents how many times the divisor can "fit" into the dividend, leading to a polynomial result. In the context of polynomial division:

• The quotient reflects a polynomial.
• This polynomial's degree will be the degree of the dividend minus the degree of the divisor.
• It's accompanied by a remainder in many cases, which ensures complete accuracy of the division.

Finding the correct quotient is crucial as it forms the bulk of the division expression, contributing directly to the solution of the polynomial division problem.