Question

Multiple Choice If \(f(x)=q(x) g(x)+r(x),\) the function \(r(x)\) is called the ________. (a) remainder (b) dividend (c) quotient (d) divisor

Short answer

The function \(r(x)\) is called the remainder.

Step-by-step solution

Understand the Division Algorithm

According to the Division Algorithm, for any two polynomials, where the divisor is non-zero, there exist unique polynomials, a quotient and a remainder, such that the original polynomial (dividend) is equal to the divisor multiplied by the quotient plus the remainder.

Identifying the Components

Given the function decomposition: \[f(x)=q(x) g(x)+r(x)\], where: - \(f(x)\) is the dividend, the polynomial being divided. - \(g(x)\) is the divisor, the polynomial by which you divide. - \(q(x)\) is the quotient, the result of the division. - \(r(x)\) is the remainder, the part of \(f(x)\) that cannot be evenly divided by \(g(x)\).

Determine the Correct Answer

From the given information and the components identified, \(r(x)\) is the remainder part of the function decomposition.

Key concepts

Division Algorithm

The Division Algorithm is a fundamental concept in polynomial arithmetic. It states that for any two polynomials, where the divisor is not zero, there are unique polynomials named quotient and remainder which satisfy the equation: \[f(x) = q(x)g(x) + r(x)\]. This can be visualized as:

• \(f(x)\): the polynomial being divided (dividend)
• \(g(x)\): the polynomial that divides \(f(x)\) (divisor)
• \(q(x)\): the result of the division (quotient)
• \(r(x)\): the leftover part (remainder)

This algorithm is similar to the long division method used in arithmetic, ensuring the dividend equals the product of the divisor and the quotient, plus the remainder.

Polynomials

Polynomials are expressions made up of variables and coefficients, linked together using addition, subtraction, and multiplication. For example, \(2x^3 + 3x^2 - x + 5\) is a polynomial. Key components include:

• Variables (e.g., \(x\))
• Coefficients (numerical factors like 2, 3, -1, 5)
• Constants (terms without variables, here, it's 5)

Polynomials can describe a wide range of mathematical and real-world phenomena. Knowing how to perform operations on them, such as addition, subtraction, multiplication, and division, is essential in higher mathematics.

Remainder

In polynomial division, the remainder \(r(x)\) is the part of the dividend that is left over after division. According to the Division Algorithm: \[ r(x) = f(x) - q(x)g(x) \]. This remainder will always have a degree less than the degree of the divisor. If the divisor has a degree of 2, the remainder could be a polynomial or a constant but never higher than degree 1. Remember, the remainder helps complete the equation, ensuring that the dividend is accurately represented by the sum of the product of the divisor and quotient, plus this leftover part.

Quotient

The quotient \(q(x)\) is the result obtained from dividing one polynomial by another. When we perform the division of \(f(x)\) by \(g(x)\), \(q(x)\) shows how many times \(g(x)\) fits into \(f(x)\). It is a crucial part of the Division Algorithm. Consider: \[ q(x) = \frac{f(x) - r(x)}{g(x)} \]. In the polynomial division process, the quotient's degree will be the difference between the degrees of the dividend and the divisor.

Dividend

The dividend \(f(x)\) is the polynomial being divided. It is the polynomial you start with in the division process. Using the Division Algorithm: \[ f(x) = q(x)g(x) + r(x) \]. The dividend holds the entire expression and is broken down into parts—the product of the divisor and quotient and the remainder. Understanding the role of the dividend helps in systematically applying division steps and verifying results.

Divisor

The divisor \(g(x)\) is the polynomial by which the dividend \(f(x)\) is divided. It's a critical factor in determining the quotient and remainder. In the equation: \[ f(x) = q(x)g(x) + r(x) \], the divisor must not be zero, as division by zero is undefined. It's important to understand that the polynomial division process heavily revolves around the divisor to achieve the correct quotient and remainder.