Question

Choose the division problem that cannot be done using synthetic division. (a) \(2 x^{3}-4 x^{2}+6 x-8\) is divided by \(x-8\) (b) \(x^{4}-3\) is divided by \(x+1\) (c) \(x^{5}+3 x^{2}-9 x+2\) is divided by \(x+10\) (d) \(x^{4}-5 x^{3}+3 x^{2}-9 x+13\) is divided by \(x^{2}+5\)

Short answer

(d) cannot be done using synthetic division because the divisor is not linear.

Step-by-step solution

Understand Synthetic Division

Synthetic division is a simplified form of polynomial division, specifically used when dividing by a linear factor of the form \(x - c\)

Identify Linear Divisors

Review each division problem to identify if the divisor is linear (of the form \(x - c\) or \(x + c\)): \(x - 8\) in (a), \(x + 1\) in (b), \(x + 10\) in (c), and \(x^{2} + 5\) in (d).

Check for Non-Linear Divisors

Identify which divisor is not linear. In this case, \(x^{2} + 5\) for (d) is not a linear factor, making it unsuitable for synthetic division.

Key concepts

Polynomial Division

Understanding polynomial division is crucial for solving division problems involving polynomials. Polynomial division is similar to long division with numbers but involves variables. There are two primary methods: long division and synthetic division. Here's what happens:

• Arrange both the dividend and the divisor in standard form, from highest to lowest degree.
• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by this quotient and subtract from the dividend.
• Repeat the process with the new polynomial that results from the subtraction.

Long division works universally, while synthetic division is a streamlined method usable only in specific cases. Understanding both methods is essential for handling various polynomial division problems.

Linear Factors

Linear factors are polynomials of degree one, represented generally as \(x - c\) or \(x + c\). A polynomial's linear factor divides it perfectly and is crucial in simplifying polynomial division. They take the form:

• \(x - c\) where \(c\) is a constant, results in a root \(c\) if substituted into the polynomial.
• \(x + c\) works similarly, here the root is \(-c\).

Linear factors are particularly beneficial in synthetic division, as this method simplifies the process by focusing on these forms. For example: to divide \(2x^3 - 4x^2 + 6x - 8\) by \(x - 8\), we'd use synthetic division because \(x - 8\) is linear.

Synthetic Division Limitations

Despite its efficiency, synthetic division has specific limitations. It can only be used in scenarios where:

• The divisor is a linear factor, i.e., of the form \(x - c\) or \(x + c\).
• It is ideally suitable for polynomials with coefficients and roots that simplify easily.

When non-linear factors such as \(x^2 + 5\) are present, synthetic division is not applicable. In such cases, you must resort to long division. For instance, dividing \(x^4-5x^3+3x^2-9x+13\) by \(x^2 + 5\) requires long division since the factor is quadratic. Always check the divisor's form to decide the appropriate method for polynomial division.