Problem 31 - Basic Technical Mathematics with Calculus

Question

Perform the indicated divisions by synthetic division. $$\left(2 x^{4}+x^{3}+3 x^{2}-1\right) \div(2 x-1)$$

Step-by-Step Solution

Answer

The quotient is \(2x^3 + 2x^2 + 4x + 2\) with no remainder.

Step 1: Set Up Synthetic Division

Identify the coefficients of the dividend polynomial, which are \(2, 1, 3, 0, -1\). Note that we added a \(0\) for the missing \(x^1\) term. The divisor is \(2x - 1\). For synthetic division, we set \(2x - 1 = 0\) to solve for \(x\), which gives \(x = \frac{1}{2}\). Use \(\frac{1}{2}\) as the synthetic division coefficient.

Step 2: Perform the Synthetic Division

Write down the coefficients \(2, 1, 3, 0, -1\). Bring the leading coefficient down: \(2\). Multiply \(2\) by \(\frac{1}{2}\), get \(1\), and add to the next coefficient \(1\), resulting in \(2\). Continue: \(2 \times \frac{1}{2} = 1\); add to \(3\), getting \(4\). Then, \(4 \times \frac{1}{2} = 2\); add to \(0\), getting \(2\). Finally, \(2 \times \frac{1}{2} = 1\); add to \(-1\), getting \(0\). So, remainder is \(0\).

Step 3: Write the Resulting Polynomial

The resulting polynomial from synthetic division is based on the coefficients from the synthetic division: \(2, 2, 4, 2\). The quotient polynomial is \(2x^3 + 2x^2 + 4x + 2\), as there is no remainder.

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another, similar to how we divide numbers. It aims to find the quotient and remainder, when a polynomial known as the dividend is divided by another polynomial called the divisor. When using polynomial division, we are often working with polynomials expressed in terms of a single variable.

• The dividend is the polynomial you are dividing, in this case, \(2x^4 + x^3 + 3x^2 - 1\).
• The divisor is the polynomial by which you divide, here it is \(2x - 1\).
• We apply synthetic division, a streamlined form of polynomial division, especially beneficial when dealing with linear divisors.

The primary goal is to reduce the degree of the dividend by taking out the divisor as many times as possible, until a remainder possibly less than the degree of the divisor is achieved.

Polynomial Coefficients

Polynomial coefficients are the numerical values that multiply the variable terms in a polynomial. They significantly determine the shape and orientation of the polynomial graph. These coefficients are part of each term in a polynomial, represented simply as numbers in the context of synthetic division.

• For our polynomial \(2x^4 + x^3 + 3x^2 - 1\), the coefficients are \(2, 1, 3, 0, -1\).
• It’s important to note the inclusion of \(0\) for the missing \(x^1\) term to maintain the polynomial's structure in synthetic division.

When we use synthetic division, we handle these coefficients directly, streamlining the division process, bypassing the manipulation of polynomials symbolically.

Divisor in Synthetic Division

The divisor in synthetic division has a crucial role in determining the values we use throughout the computation. In our problem, the divisor is \(2x - 1\). To use synthetic division, we need a value derived from the divisor that will guide our operation.

• The process starts with setting the divisor \(2x - 1\) equal to zero: \(2x - 1 = 0\), which simplifies to \(x = \frac{1}{2}\).
• In synthetic division, \(\frac{1}{2}\) is referred to as the synthetic division coefficient or root.

This coefficient is systematically used to multiply with current terms and add to the subsequent coefficient, effectively reducing the polynomial's degree and leading us towards the quotient.