Problem 33 - College Algebra

Question

For the following exercises, use synthetic division to find the quotient. $$ \left(x^{4}-8 x^{3}+24 x^{2}-32 x+16\right) \div(x-2) $$

Step-by-Step Solution

Answer

The quotient is \(x^3 - 6x^2 + 12x - 8\).

Step 1: Prepare for Synthetic Division

Synthetic division is a simpler way to divide polynomials compared to the traditional long division method. In synthetic division, we only use the coefficients of the polynomial. Here, the divisor is \(x-2\), so we take \(x = 2\) as our value to use in synthetic division. List the coefficients of the dividend \(x^4 - 8x^3 + 24x^2 - 32x + 16\), which are \(1, -8, 24, -32, 16\).

Step 2: Set Up the Synthetic Division Table

Draw a horizontal line, and below it write the coefficients of the polynomial: \(1, -8, 24, -32, 16\). To the left of this line, write the value \(2\), which is derived from the divisor \(x-2\).

Step 3: Conduct First Synthetic Division Step

Bring down the first coefficient, which is \(1\). Multiply \(1\) by \(2\), getting \(2\), and place it under the second coefficient \(-8\). Then, add \(-8\) and \(2\) to get \(-6\).

Step 4: Continue Synthetic Division Process

Multiply the new result \(-6\) by \(2\) to get \(-12\). Place this under the third coefficient \(24\) and add them to get \(12\). Repeat by multiplying \(12\) by \(2\) to get \(24\), place it under \(-32\), and add to get \(-8\). Then, multiply \(-8\) by \(2\) to get \(-16\), place it under \(16\), and add to get \(0\).

Step 5: Write the Quotient

The final numbers below the line, \(1, -6, 12, -8\), represent the coefficients of the resulting polynomial, which is of degree one less than the original polynomial. Thus, the quotient is \(x^3 - 6x^2 + 12x - 8\).

Step 6: Verify the Remainder

Since the last number in the synthetic division process is \(0\), this confirms that \(x-2\) is a factor of \(x^4 - 8x^3 + 24x^2 - 32x + 16\) and the division is exact with no remainder.

Key Concepts

Polynomial Division

Polynomial division is a mathematical technique used to divide one polynomial by another. It's similar to long division with numbers, but involves variables and exponents. When performing polynomial division, we usually deal with expressions like \(f(x) \div g(x)\), where \(f(x)\) is the dividend and \(g(x)\) is the divisor.

The method of synthetic division is commonly used when dividing polynomials by a linear factor. This is often faster and less error-prone than long division. It simplifies calculations by focusing only on coefficients, ignoring variables during the process.

• Identify the divisor, e.g. \(x - a\), and use \(a\) for synthetic division.
• List the coefficients of the dividend polynomial.
• Set up a synthetic division table and perform the operations step by step.

Synthetic division provides an efficient way to find the quotient and remainder, particularly useful for higher-degree polynomials.

Quotient of Polynomials

The quotient of polynomials is the result you get after dividing one polynomial by another. In synthetic division, the quotient is constructed from the numbers below the line in the last synthetic division table.

When you divide, you reduce the degree of the polynomial, meaning the resulting quotient is of a lower degree than the original polynomial. For example, if you start with a polynomial of degree 4 and divide it by a linear polynomial (degree 1), the quotient will be of degree 3.

• The quotient coefficients form a new polynomial with one degree less than the original dividend.
• In the mentioned example, the quotient polynomial is \(x^3 - 6x^2 + 12x - 8\).

Understanding the quotient is crucial, as it helps illustrate the polynomial's behavior and its division accuracy.

Factors of Polynomials

Factors of polynomials are divisors that break down the polynomial into simpler components or smaller degree polynomials. If a polynomial can be divided by a factor without leaving any remainder, the factor is valid. In synthetic division, if the remainder is zero, it indicates that the divisor is a factor.

This is particularly useful for simplifying polynomials and finding the roots or zeros. In the exercise above, since the remainder is zero, \(x-2\) is a confirmed factor of \(x^4 - 8x^3 + 24x^2 - 32x + 16\).

• Factors help in solving polynomial equations.
• Finding factors can aid in graphing polynomial functions.
• They play a critical role in determining polynomial roots.

By understanding polynomial factors, you gain insights into polynomial behavior and solutions.