Problem 9 - 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 the quotient \(P(x) / D(x)\) in the form $$\frac{P(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$$ $$P(x)=x^{2}+4 x-8, \quad D(x)=x+3$$

Step-by-Step Solution

Answer

The quotient is \(x + 1\) with a remainder of \(-11\).

Step 1: Setup the Synthetic Division

For synthetic division, we first need the opposite of the constant term from the divisor \((x + 3)\), which is \(-3\). Write \(-3\) aside the coefficients of the dividend \(P(x) = x^2 + 4x - 8\), which are \(1, 4, -8\).

Step 2: Perform Synthetic Division

Bring down the first coefficient (1) directly. Multiply \(-3\) by 1 to get \(-3\), and add this to the next coefficient (4) to get \(1\). Repeat: \(-3\times1 = -3\); add to \(-8\) to get \(-11\). The quotient is \(x + 1\), and the remainder is \(-11\).

Step 3: Express the Result

Rewrite the division result accordingly. The quotient \(Q(x)\) is \(x + 1\) and the remainder \(R(x)\) is \(-11\). Thus, \[\frac{P(x)}{D(x)} = (x + 1) + \frac{-11}{x+3}.\]

Key Concepts

Synthetic Division

Synthetic division is a streamlined method for dividing polynomials when the divisor is a linear factor of the form \(x + c\). It simplifies calculations by focusing only on coefficients and involves fewer steps than traditional long division.
To perform synthetic division, you need:

• The coefficients of the dividend polynomial, \(P(x)\).
• The opposite of the constant term in the linear divisor \(x + c\).

Write this opposite number to the left of a row of coefficients from \(P(x)\). Begin by bringing down the first coefficient as is. Multiply it by the number written aside to generate new terms, which you add successively to the other coefficients.
This method quickly gives you the quotient polynomial and any remainder, making synthetic division a quick option for finding division results.

Long Division in Algebra

Long division in algebra is similar to numerical long division, but it applies to polynomials. This process is not as quick as synthetic division but is powerful for any type of polynomial division, not just those with linear divisors.
Here’s how you perform long division with polynomials:

• Divide the first term of the dividend by the first term of the divisor.
• Multiply the entire divisor by this result and subtract it from the original dividend.
• Repeat the process with the new polynomial left from the subtraction.

Continue until the degree of the remainder is less than the degree of the divisor. Long division provides a clear route to find both the quotient and remainder, allowing any division to be expressed precisely.

Quotient and Remainder

In polynomial division, the quotient and remainder are essential outcomes.
The quotient, \(Q(x)\), is the polynomial obtained after dividing the terms of the dividend by the divisor, and it's of lower degree than the dividend. Meanwhile, the remainder, \(R(x)\), is the leftover part after the division completes, with a degree less than that of the divisor.
The Division Algorithm for polynomials states that any polynomial \(P(x)\) divided by another polynomial \(D(x)\) results in:

• A quotient polynomial \(Q(x)\)
• A remainder polynomial \(R(x)\)

such that \(P(x) = D(x) \cdot Q(x) + R(x)\). This relation allows any division to be expressed as a quotient with an added fractional part, offering clear insight into the division results.

Polynomial Expressions

Polynomial expressions consist of variables raised to whole number exponents, combined using addition, subtraction, and multiplication. They form the basis of more complex algebraic manipulations such as polynomial division.
Polynomials are classified by their degree, which determines the methods available for division:

• Linear polynomials, such as \(x + c\), are ideal for synthetic division.
• Higher-degree polynomials require approaches like long division.

Understanding the structure of polynomial expressions helps in determining the most suitable division technique. This awareness facilitates breaking down complex polynomials into simpler components, making calculations more manageable and results clearer.