Problem 29 - College Algebra with Corequisite Support

Question

For the following exercises, use synthetic division to find the quotient. Ensure the equation is in the form required by synthetic division. (Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor.) $$ \left(x^{4}+x^{3}-3 x^{2}-2 x+1\right) \div(x+1) $$

Step-by-Step Solution

Answer

The quotient is \(x^3 - 3x - 2\).

Step 1: Identify Coefficients of Dividend

The first step in synthetic division is to identify the coefficients of the polynomial you need to divide (the dividend). For the polynomial \(x^4 + x^3 - 3x^2 - 2x + 1\), the coefficients are \([1, 1, -3, -2, 1]\).

Step 2: Set Up the Division

Since the divisor is \(x + 1\), we set up synthetic division by using the zero of \(x + 1 = 0\), which is \(-1\). Write \(-1\) to the left of the coefficients.

Step 3: Bring Down the Leading Coefficient

Begin the division process by bringing down the first coefficient (1) directly to the bottom row.

Step 4: Perform the Division

Multiply the number brought down (1) by \(-1\) and write the result under the next coefficient (1). Add these to get a new coefficient value, which is \(0\). Repeat this process for all coefficients: - Next column: Multiply \(0\) by \(-1\), put under \(-3\), sum is \(-3\).- Next: Multiply \(-3\) by \(-1\), put under \(-2\), sum is \(1\).- Next: Multiply \(1\) by \(-1\), put under \(1\), sum is \(0\).

Step 5: Interpret the Results

The numbers on the bottom row are the coefficients of the quotient polynomial, starting from one degree less than the original polynomial. The quotient is \(x^3 - 3x - 2\) with a remainder of 0.

Key Concepts

Polynomial Division

Polynomial division is a technique used to divide one polynomial by another. It's much like long division with numbers. In this context, synthetic division offers a simpler way to divide polynomials, especially when the divisor is a linear polynomial. The goal is to separate a higher degree polynomial (the dividend) by a lower or same degree polynomial (the divisor). This process yields a quotient (result of the division) and possibly a remainder if the division isn't exact. For the given polynomial division problem \[x^4 + x^3 - 3x^2 - 2x + 1\]divided by \[x + 1\],synthetic division provides a streamlined method to find the quotient without having to perform complex algebraic manipulations.

Coefficients

At the heart of polynomial operations like synthetic division are coefficients. Coefficients in a polynomial are the numerical factors accompanying each term. For instance, in the polynomial \[x^4 + x^3 - 3x^2 - 2x + 1\],which we are dividing, the coefficients are \([1, 1, -3, -2, 1]\).In synthetic division, these coefficients are arranged in a specific order to facilitate the division.

• The sequence starts from the coefficient of the highest power term.
• Consistent arrangement is crucial to ensure accurate calculations.

This setup precedes the iterative process where these coefficients are gradually transformed into those of the quotient polynomial.

Quotient and Remainder

The results of polynomial division are expressed in terms of a quotient and possibly a remainder. The quotient is a polynomial of lesser degree than the dividend, representing the main result of the division. In our specific example, the polynomial division of \[x^4 + x^3 - 3x^2 - 2x + 1\]by \[x + 1\]produces a quotient \[x^3 - 3x - 2\]with a remainder of 0.

• The quotient showcases the factorization of the original polynomial over the divisor.
• The remainder, when zero, indicates the divisor divides the dividend exactly without leftover terms.

Thus, the absence of a remainder simplifies the solution, emphasizing an exact division.

Division Algorithm for Polynomials

The division algorithm for polynomials is a fundamental concept that guarantees any polynomial \((P(x))\)can be expressed as \[P(x) = D(x)Q(x) + R(x)\],where \(D(x)\)is the divisor, \(Q(x)\)is the quotient, and \(R(x)\)is the remainder. This foundational algorithm offers insight into how polynomials relate to each other through division.

• When the remainder \(R(x)\)is zero or has a lesser degree than the divisor, the algorithm further validates the results.
• The algorithm aids in polynomial factorization, where the quotient becomes a new factor of the polynomial.

In synthetic division, this algorithm underpins the swift and efficient calculation of results, streamlining the division of polynomials.