Problem 5 - Precalculus Mathematics for Calculus

Question

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 \(P(x)=D(x) \cdot Q(x)+R(x).\) $$P(x)=2 x^{3}-3 x^{2}-2 x, \quad D(x)=2 x-3$$

Step-by-Step Solution

Answer

Divide to find \( Q(x) = x^2 - 1 \) and remainder \( R(x) = 3 \). Thus, \( P(x) = (2x-3)(x^2-1)+3 \).

Step 1: Identify Dividend and Divisor

Here, the polynomial to be divided is our dividend, denoted by \( P(x) = 2x^3 - 3x^2 - 2x \), and the divisor is \( D(x) = 2x - 3 \). We will divide \( P(x) \) using \( D(x) \).

Step 2: Set Up Long Division

For long division, we first write \( P(x) \) inside the division symbol and \( D(x) \) outside. We start by taking the leading term of \( P(x) \), which is \( 2x^3 \), and divide it by the leading term of \( D(x) \), which is \( 2x \).

Step 3: Divide Leading Terms

The leading term \( 2x^3 \) divided by \( 2x \) gives \( x^2 \). This \( x^2 \) becomes the first term of the quotient polynomial \( Q(x) \). Write \( x^2 \) above the division line.

Step 4: Multiply and Subtract First Term

Multiply \( x^2 \) by each term of \( D(x) \), yielding \( 2x^3 - 3x^2 \). Subtract this from \( P(x) \), which is \( 2x^3 - 3x^2 - 2x \), to obtain \( 0x^3 - 3x^2 - 2x - (-3x^2) \), resulting in \( 0 - (-3x^2 + 2x) = 0x^2 - 2x \).

Step 5: Repeat Process

Next, divide \( -2x \) by \( 2x \), which gives \(-1\). Write \(-1\) as the next term in \( Q(x) \). Multiply \(-1\) by \( D(x) \), giving \(- (2x - 3) = -2x + 3\). Subtract from the result \( 0x^2 - 2x \) to get \( 0x + 3\).

Step 6: Write Remainder

Since \( 3 \) cannot be further divided by \( 2x - 3 \), it is the remainder \( R(x) \). Now, \( P(x) \) can be expressed in the required form: \( P(x) = (2x - 3)(x^2 - 1) + 3 \).

Step 7: Check the Final Expression

The expression \( P(x) = D(x) \cdot Q(x) + R(x) \) should simplify back to the original dividend if calculations are correct. Substitute to verify: \( (2x - 3)(x^2 - 1) + 3 = 2x^3 - 3x^2 - 2x \).

Key Concepts

Synthetic Division

Synthetic division is a shorthand method for dividing a polynomial by a linear divisor of the form \(x - c\). It's quicker and involves less writing than long division. Although it’s not used for divisors with a degree higher than one, it’s a quick alternative when applicable.
To perform synthetic division, follow these steps:

• Write down the coefficients of the polynomial \(P(x)\).
• Use the value of \(c\) from the divisor \(D(x) = x - c\).
• Bring the first coefficient straight down as is.
• Multiply this brought down number by \(c\) and write the result under the next coefficient.
• Add the numbers in the column and repeat the process until you reach the last coefficient.

The final row gives the coefficients of the quotient polynomial, and if any number remains, it acts as the remainder.
Synthetic division works best for simpler polynomials where the divisor is easily extractable as \(x - c\).

Long Division

Long division of polynomials is much like long division in arithmetic. It's a methodical approach that breaks down a tougher polynomial division. It encompasses these steps:

• Write the dividend \(P(x)\) and the divisor \(D(x)\) appropriately.
• Divide the leading term of \(P(x)\) by the leading term of \(D(x)\) to find the first term of the quotient \(Q(x)\).
• Multiply this term by \(D(x)\) and subtract the result from \(P(x)\) to obtain a new polynomial.
• Repeat this process with this new polynomial until the degree of the remainder is less than that of the divisor.

Throughout these steps, keep arranging and calculating carefully. It’s a thorough approach but ensures clarity when synthetic division is not applicable or might confuse.

Remainder Theorem

The remainder theorem provides a quick way to find the remainder of a polynomial division without performing the entire division process. It states that if a polynomial \(P(x)\) is divided by \(x-c\), the remainder is \(P(c)\).
This theorem helps verify the results of polynomial division:

• Substitute \(c\) in the polynomial \(P(x)\).
• Evaluate \(P(c)\).

The obtained value is the remainder of the division.
Using this theorem can save time, especially during calculations or while checking one's work post-division.

Quotient Polynomial

Upon dividing two polynomials, the quotient polynomial is the result without the remainder. It's written as \(Q(x)\), representing the factor left after division.
The process of finding \(Q(x)\) involves:

• Accurately setting up either synthetic or long division.
• Following through with the division steps to systematically reduce the problem.
• Collecting the terms as they're produced during the division process.

After achieving \(Q(x)\), one verifies correctness through multiplication, ensuring the original polynomial can be recreated by: \(P(x) = D(x) \cdot Q(x) + R(x)\).
This step ensures the reliability of one's solution, confirming the quotient and remainder.