Problem 12 - Basic Technical Mathematics with Calculus

Question

Find the remainder using the remainder theorem. Do not use synthetic division. $$\left(4 x^{4}-x^{3}+5 x-7\right) \div(x-5)$$

Step-by-Step Solution

Answer

The remainder is 2393.

Step 1: Identify Polynomial and Divisor

The given polynomial is \(4x^4 - x^3 + 5x - 7\) and the divisor is \(x - 5\). According to the Remainder Theorem, the remainder of the division of a polynomial \(f(x)\) by a linear divisor \((x - a)\) is the value of \(f(a)\). Here, \(a\) is 5.

Step 2: Substitute "a" into the Polynomial

Substitute \(x = 5\) into the polynomial \(f(x) = 4x^4 - x^3 + 5x - 7\).

Step 3: Evaluate the Polynomial at x = 5

Calculate each term: - First term: \(4(5)^4 = 4 \times 625 = 2500\)- Second term: \(-(5)^3 = -125\)- Third term: \(5(5) = 25\)- Fourth term: \(-7\)Add these up to find \(f(5)\).

Step 4: Calculate the Result

Add the values from the previous step: \(2500 - 125 + 25 - 7 = 2393\).

Step 5: State the Remainder

The remainder when \(4x^4 - x^3 + 5x - 7\) is divided by \(x - 5\) is 2393.

Key Concepts

Polynomial Division

Polynomial division is an arithmetic operation similar to integer division, applied to polynomials. It breaks a polynomial dividend by a divisor to get a quotient and a remainder. The aim is to see how many times the divisor fits into the dividend and what is left over. This process is common in algebra and calculus, helping to simplify complex polynomial expressions.

When performing polynomial division, the order of the terms and their alignment is crucial. Typically, you write the polynomial in descending order of the powers of the variable, which makes it easier to perform the operation. In some exercises, such as those using the Remainder Theorem, you might not perform traditional long division, but understanding this concept prepares you for more comprehensive algebraic manipulations.

Linear Divisor

A linear divisor is a polynomial of degree one, generally expressed in the form \(x - a\) where \(a\) is a constant. In the context of polynomial division, when you have a linear divisor, it simplifies the process as per the Remainder Theorem, which streamlines finding the remainder after division without full division computation.

The linear divisor is essential in many algebraic operations because it sets the foundation for solving equations and inequalities. In problems, like the one presented, \(x - 5\) is a linear divisor. The constant term \(5\) indicates where to evaluate the polynomial to find the remainder efficiently.

Evaluate Polynomial

Evaluating a polynomial involves replacing the variable, usually \(x\), with a specific number and solving to find the result. This process requires careful arithmetic operations, especially handling exponents and coefficients.

Using the exercise as an example, given the polynomial \(4x^4 - x^3 + 5x - 7\), you substitute \(x = 5\), as directed by the Remainder Theorem. Calculating each term, you evaluate \(4(5)^4\), \(-(5)^3\), and so on, step by step until the sum gives you the result. Practicing this evaluation helps build confidence and accuracy in solving polynomial functions, a critical skill in higher mathematics.

Mathematics Education

Mathematics education aims to build a strong foundation in concepts like polynomial division and the Remainder Theorem. These topics help students understand more complex algebraic operations that are useful in advanced studies and everyday problem-solving.

Educators emphasize clarity, practice, and step-by-step problem-solving approaches. By gradually guiding students through examples and offering problem-solving techniques, like using the Remainder Theorem, students develop critical thinking and logical reasoning. This nurturing environment encourages a deeper understanding of mathematics, equipping students with essential skills for academic and professional success.