Problem 27 - Basic Technical Mathematics with Calculus

Question

Perform the indicated divisions by synthetic division. $$\left(p^{6}-6 p^{3}-2 p^{2}-6\right) \div(p-2)$$

Step-by-Step Solution

Answer

The quotient is \(p^5 + 2p^4 + 4p^3 + 2p^2 - 2p - 4\) with a remainder of \(-14\).

Step 1: Set Up the Synthetic Division

Start with the divisor, which is \(p - 2\). We use the zero of this term, \(p = 2\), for the synthetic division. Next, write down the coefficients of the dividend \(p^6 - 6p^3 - 2p^2 - 6\). Remember that we need to include all terms down to the constant, so fill in any missing degrees with zeros: \([1, 0, 0, -6, -2, 0, -6]\).

Step 2: Perform Synthetic Division

Write the zero of the divisor (2) to the left. Bring down the leading coefficient \(1\) of the dividend to below the division line. Multiply \(1\) by \(2\), place the result \(2\) under the next coefficient (0), add them, and write the result \(2\) below. Continue this process: multiply the new result by 2 and add down the row.

Step 3: Continue the Process

Following the same process, multiply the sum by 2, add to the next coefficient, and repeat. The values should line up as: \([1, 0, 0, -6, -2, 0, -6] \to [1, 2, 4, 2, -2, -4, -14]\).

Step 4: Interpret the Result

The final row, excluding the last number, represents the coefficients of the quotient \(1, 2, 4, 2, -2, -4\). Hence, the quotient is \(p^5 + 2p^4 + 4p^3 + 2p^2 - 2p - 4\), and \(-14\) is the remainder.

Key Concepts

Polynomial Division

Polynomial division is a method used to divide polynomials, similar to how long division is used with numbers. It's particularly useful for dividing a polynomial by a binomial.

• When dividing polynomials, the goal is to determine how many times the divisor polynomial goes into the dividend polynomial, much like in numerical division.
• This process can be simplified using synthetic division, which requires fewer calculations and is faster for divisors of the form \((p-k)\).

Make sure to align all-like degree terms when setting up your division to avoid mistakes.

Remainder Theorem

The remainder theorem is a principle that connects the remainder of a polynomial division to the polynomial's evaluation at a certain point. It states:

• If a polynomial \(f(p)\) is divided by \((p - k)\), the remainder is \(f(k)\).
• This means in our exercise, to find the remainder when dividing by \((p - 2)\), we evaluate the polynomial at \(p=2\).
• If the polynomial is expressed as \(f(p) = p^6 - 6p^3 - 2p^2 - 6\), the remainder can be verified by calculating \(f(2)\), which should also yield \(-14\).

The theorem provides a quick way to check your division work.

Coefficients

Coefficients are the numbers that multiply the variables in a polynomial. In synthetic division, the focus is on these coefficients:

• Typically, you will list the coefficients of the dividend polynomial, including zeros for any missing terms, to ensure a comprehensive division process.
• Each step in synthetic division involves arithmetic operations on these coefficients: extending downward multiplication and summation sequences.
• For the polynomial \(p^6 - 6p^3 - 2p^2 - 6\), the coefficients are recorded as \([1, 0, 0, -6, -2, 0, -6]\), maintaining the integrity of each polynomial degree.

Getting the coefficients right is critical for a successful division.

Divisor and Dividend in Algebra

In algebraic division, it's essential to understand the roles of the divisor and the dividend:

• The dividend is the polynomial you're dividing into, which in this exercise is \(p^6 - 6p^3 - 2p^2 - 6\).
• The divisor, here \((p - 2)\), is what you divide by, and its zero is \(p=2\), used in synthetic division.
• It's crucial to prepare both the divisor and dividend correctly to perform the division accurately.

Keenly organizing your setup helps avoid confusion and ensures an accurate quotient and remainder.