Problem 8 - Algebra and Trigonometry with Analytic Geometry

Question

Find the quotient and remainder if \(f(x)\) is divided by \(p(x)\). $$ f(x)=7 x^{2}+3 x-10 ; \quad \quad p(x)=x^{2}-x+10 $$

Step-by-Step Solution

Answer

Quotient is 7, remainder is \(10x - 80\).

Step 1: Set up the division

We will perform polynomial long division using \(f(x) = 7x^2 + 3x - 10\) and \(p(x) = x^2 - x + 10\). First, determine how many times the leading term of \(p(x)\) goes into the leading term of \(f(x)\). Here, divide \(7x^2\) by \(x^2\) to get \(7\).

Step 2: Multiply and subtract

Multiply the entire polynomial \(p(x)\) by the quotient term found in Step 1, which is \(7\), resulting in \(7(x^2 - x + 10) = 7x^2 - 7x + 70\). Subtract this from \(f(x)\): \(7x^2 + 3x - 10 - (7x^2 - 7x + 70)\).

Step 3: Simplify the subtraction

Simplify the result from Step 2: \(7x^2 + 3x - 10 - 7x^2 + 7x - 70 = 10x - 80\). This is the remainder after dividing.

Key Concepts

Understanding the Quotient in Polynomial Division

When you divide two polynomials, one key piece of information you find is the *quotient*. The quotient is the result of the division, similar to how in regular numbers, dividing 8 by 4 gives a quotient of 2. In our original exercise, dividing the polynomial \(f(x) = 7x^2 + 3x - 10\) by \(p(x) = x^2 - x + 10\) gives a quotient of 7.

To find this, you look at the leading term of both polynomials. You divide the leading term of the dividend, \(f(x)\), by the leading term of the divisor, \(p(x)\). Here, \(7x^2\) divided by \(x^2\) equals 7. This number is placed at the top of your division setup, indicating how many times the divisor fits into the dividend at this step.

The quotient can change depending on polynomial complexity, and sometimes, you may need more than one term in your quotient. But in this example, a single term of 7 is sufficient. Even if the remainder isn't zero, the quotient tells you how the larger polynomial breaks down in terms of the smaller one.

What is the Remainder in Polynomial Division?

The remainder in polynomial division works like the remainder in arithmetic division. It's what's left over after dividing the polynomials as much as possible. In this exercise, after finding our quotient of 7, we subtract the product of this quotient and the divisor from the original polynomial \(f(x)\).

To elaborate, after multiplying \(7\) by \(x^2 - x + 10\), we get \(7x^2 - 7x + 70\). Subtracting this from \(f(x)\), we calculate: \(7x^2 + 3x - 10 - (7x^2 - 7x + 70)\). Simplifying this subtraction results in a remainder of \(10x - 80\).

The remainder is crucial because it tells us how close our division is to being perfect. If the remainder is zero, it indicates that the divisor is an exact factor of the dividend. But in many cases, such as this one, the remainder will not be zero, showing there's a part of the dividend that doesn't perfectly divide.

Polynomials and Their Structure

A polynomial is an expression made up of variables, coefficients, and constants, connected through operations of addition, subtraction, and multiplication. The variables in a polynomial are typically represented with powers, such as \(x^2\), indicating squared terms.

In our problem, polynomial \(f(x) = 7x^2 + 3x - 10\) is being divided by the polynomial \(p(x) = x^2 - x + 10\). Both of these have structures that dictate how they can interact mathematically. The degrees of each term (the highest powers of the variables) are important when performing division, as they determine how the leading terms are utilized.

Understanding polynomials is crucial for manipulation and division. Every term serves a purpose in defining the polynomial's behavior and characteristics. Recognizing the highest degree or leading term allows for initial steps in division and many other forms of algebraic manipulation.

Connecting Polynomial Division to Algebra

Polynomial division is a specific technique used in algebra, providing insight into factorization and simplifying algebraic expressions. It's a form of algebraic manipulation that simplifies more complex polynomials into unreduced forms.

In algebra, these skills are necessary to solve many forms of equations, analyze functions, and understand polynomial expressions' behavior. Solving for the intersection points between two polynomial functions, for instance, often involves division and its competencies.

Additionally, grasping polynomial division deepens your understanding of how algebraic expressions work together. It acts as a foundation for higher studies in algebra, factoring, and the simplification of rational expressions, making it an essential skill for future mathematical proficiency.