Problem 28 - Algebra and Trigonometry

Question

25-38 . Find the quotient and remainder using synthetic division. $$ \frac{4 x^{2}-3}{x+5} $$

Step-by-Step Solution

Answer

The quotient is \(4x - 20\) and the remainder is 97.

Step 1: Set up synthetic division

First, organize the polynomial and the divisor for synthetic division. We have the polynomial \(4x^2 + 0x - 3\) and divisor \(x + 5\). In synthetic division, we use the root of the divisor \((x + 5)\), which is \(-5\). Write the coefficients of the polynomial \([4, 0, -3]\) beside the root \(-5\).

Step 2: Bring down the leading coefficient

Bring the first coefficient (4) straight down below the line. This is now the initial part of our quotient.

Step 3: Perform synthetic multiplication and addition

Multiply the number just written below the line (4) by \(-5\) (the root), resulting in \(-20\). Write \(-20\) below the next coefficient (0). Add this to 0, the result is \(-20\). Write \(-20\) below the line.

Step 4: Repeat multiplication and addition

Again, multiply the number below the line (\(-20\)) by \(-5\), giving 100. Write 100 below the next coefficient (\(-3\)). Add this to \(-3\), resulting in 97. Write 97 below the line.

Step 5: Interpret the result

The numbers below the line represent the coefficients of the quotient and the remainder. The quotient is \(4x - 20\), and the remainder is 97. Therefore, \(\frac{4x^2 - 3}{x + 5} = 4x - 20 + \frac{97}{x+5}\).

Key Concepts

Polynomial Division

Polynomial division is a mathematical technique used to divide two polynomials. It is quite similar to long division with numbers, but with algebraic expressions instead. In our exercise, we're dividing the polynomial \(4x^2 - 3\) by \(x + 5\). There are several methods of polynomial division, but synthetic division is often preferred for its simplicity, especially when the divisor is a linear polynomial like \(x + 5\).
The main goal of polynomial division is to find the quotient (the result of division) and the remainder (what's left over if it doesn't divide perfectly). This process is especially important in algebra for simplifying expressions and solving equations. It has practical applications in calculus, algebraic number theory, and in dealing with complex polynomials in engineering.

Remainder Theorem

The Remainder Theorem is a useful principle in algebra that simplifies finding out the remainder of a polynomial division. According to the theorem, when a polynomial \(f(x)\) is divided by a linear divisor \(x - c\), the remainder of this division is simply \(f(c)\). This is because evaluating the polynomial at \(c\) gives the remainder without completing the entire polynomial division.

In our exercise, we divided \(4x^2 - 3\) by \(x + 5\) and found the remainder to be 97. By using the Remainder Theorem, we theoretically evaluate \(4(-5)^2 - 3\) to verify that it indeed provides us with 97. This characteristic makes the Remainder Theorem a powerful tool for evaluating polynomials and simplifying complex algebraic tasks.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like \(x, y\)), and arithmetic operations (like addition, subtraction, multiplication, and division). In our exercise, the expression \(4x^2 - 3\) represents a quadratic polynomial where the highest degree of \(x\) is 2. The division by \(x + 5\) involves working within such expressions to simplify or solve them.

Understanding algebraic expressions is crucial since they are foundational to many aspects of mathematics. They allow you to represent real-world situations mathematically and are used extensively in all areas of mathematics, from simple algebra to complex calculus.
Through the practice of manipulating algebraic expressions, students can learn how to simplify problems, identify patterns, and solve equations with greater ease.

Roots of Polynomials

The roots of a polynomial are the values of the variable that make the polynomial equal to zero. For instance, if you have the polynomial \(f(x) = x^2 - 3x + 2\), the roots would be the values of \(x\) that satisfy \(f(x) = 0\). In the context of division by \(x + 5\), we are searching for the root of the divisor, which is \(-5\) since setting \(x + 5 = 0\) solves to \(x = -5\).

Finding roots is a critical aspect of algebra because it provides essential information about the behavior of the polynomial function. This is often tied into synthetic division and the remainder theorem, where identifying the root helps set up the division and interpret the results properly. Understanding how to find and use roots allows students to solve polynomial equations and factor polynomials, which are both key skills in algebra and calculus.