Problem 6 - Precalculus: Functions and Graphs

Question

Find the quotient and remainder if \(f(x)\) is divided by \(p(x)\). $$f(x)=-5 x^{2}+3 ; \quad p(x)=x^{3}-3 x+9$$

Step-by-Step Solution

Answer

Quotient: 0, Remainder: \(-5x^2 + 3\).

Step 1: Identify Degrees

First, identify the degrees of the polynomials. The degree of the dividend polynomial, \( f(x) = -5x^2 + 3 \), is 2. The divisor polynomial, \( p(x) = x^3 - 3x + 9 \), has a degree of 3.

Step 2: Consider Polynomial Division

Since the degree of \( f(x) \) is lower than the degree of \( p(x) \), the division will result in a quotient of zero, \( q(x) = 0 \), and the remainder will be \( f(x) \) itself.

Step 3: Quotient Determination

With a lower degree in the dividend compared to the divisor, conclude that the quotient of the division is zero, \( q(x) = 0 \).

Step 4: Remainder Identification

The remainder of the division, due to the lower degree of the dividend, is the dividend itself: \( r(x) = f(x) = -5x^2 + 3 \).

Key Concepts

Quotient and Remainder

When dividing one polynomial by another, the result comprises two main components: the quotient and the remainder. Think of it like the traditional division of numbers, where the quotient is the result showing how many times the divisor fits into the dividend, and the remainder is what's left over. In polynomial division, when the degree of the divisor is greater than the degree of the dividend, the divisor cannot "fit" into the dividend even once, leading to a quotient of zero. The remainder, therefore, is the whole of the original dividend polynomial.

For example, in the exercise given, we have a dividend polynomial,

• Dividend: \( f(x) = -5x^2 + 3 \)
• Divisor: \( p(x) = x^3 - 3x + 9 \)

Where the dividend's degree is 2, and the divisor's degree is 3. Thus, the quotient is zero, \( q(x) = 0 \), and the remainder is the dividend itself, \( r(x) = -5x^2 + 3 \). Recognizing these situations saves time and simplifies the process of polynomial division.

Degree of Polynomials

The degree of a polynomial is a key concept in understanding how division between polynomials works. It is defined as the highest power of the variable in the polynomial. In a polynomial, expressed generally as \( ax^n + bx^{n-1} + \, \ldots + cx + d \), where 'a', 'b', 'c', and 'd' are constants, the degree is 'n' if \( ax^n \) is the highest term.

Understanding the degree of polynomials enables you to quickly determine key aspects of polynomial operations like addition, subtraction, multiplication, and division. In division particularly, the degree of the dividend and divisor directly influences the degree of the quotient. If the dividend's degree is less than the divisor's degree, the quotient is always zero. This was exactly the case in our exercise, with \( f(x) \) having a degree of 2 and \( p(x) \) having a degree of 3, indicating immediately that no full divisions are possible, so the quotient is zero.

Always check the degrees first when faced with polynomial division. This saves time and identifies whether a division will lead to further calculations or conclude with a straightforward answer.

Polynomial Long Division

Polynomial long division is a methodical way to divide polynomials that works a lot like long division with numbers. The principal goal is to divide the dividend by the divisor and determine how many times the divisor can "go into" or "fit into" the dividend, guided by the degrees of the polynomials.

Here's a step-by-step approach:

• First, write down both the dividend and the divisor.
• Compare their leading coefficients and degrees to guide each step of the division.
• Subtract products from the dividend, just like with numeric long division, and bring down the next terms.
• Repeat the process until no further division is possible.

In some cases, like the one in our exercise, polynomial long division is halted right off the bat when the dividend's degree is less than the divisor's. This initial degree check can prevent pointless calculations and confirm fast that the quotient is zero.

Despite seeming complex initially, mastering polynomial long division helps demystify challenging algebra problems and develops deeper mathematical understanding.