# Calculator Techniques FOR BOARD EXAM - Evaluating Functions and Simplifying Expressions 10 EXAMPLES

TLDRThis video tutorial introduces calculator techniques essential for students preparing for engineering board exams, covering various fields like mechanical, electrical, and electronics engineering. The presenter shares methods to evaluate functions, simplify expressions, and apply the Remainder and Factor Theorems using a calculator, focusing on multiple-choice questions. The video includes step-by-step demonstrations for ten algebraic problems, emphasizing the importance of calculator skills in passing the math section of board exams.

### Takeaways

- ๐ The video is aimed at students preparing for engineering board exams, focusing on calculator techniques for math problems.
- ๐ข The presenter introduces calculator techniques as essential for passing the math section of board exams, particularly for multiple-choice questions.
- ๐ The first topic covered is algebra, specifically evaluating functions and simplifying expressions, which is fundamental for other math topics.
- ๐ฑ The use of a calculator, specifically the CASIO fx 570, is demonstrated for evaluating functions by substituting values into the function.
- ๐ The script provides step-by-step instructions on how to input mathematical expressions into a calculator for evaluation.
- ๐ The Remainder Theorem is discussed, explaining how to find the remainder of a polynomial division using the calculator by evaluating the function at specific points.
- ๐ The Factor Theorem is also explained, showing how to determine if an expression is a factor of a polynomial by checking for a remainder of zero.
- ๐ The video script includes practical examples of how to use a calculator to solve problems related to evaluating functions, polynomial division, and factorization.
- ๐ The presenter suggests using the calculator for trial and error when factoring is difficult, especially when multiple choices are provided.
- ๐ The script emphasizes the importance of mastering calculator techniques for efficiency in solving mathematical problems during exams.
- ๐ค The video concludes with an encouragement to share the knowledge of calculator techniques with others preparing for board exams.

### Q & A

### What is the main purpose of the video 'Calculator Techniques FOR BOARD EXAM - Evaluating Functions and Simplifying Expressions'?

-The main purpose of the video is to teach calculator techniques that are useful for students preparing for board exams, particularly in engineering fields, to help them effectively solve mathematical problems that involve evaluating functions and simplifying expressions.

### Why are calculator techniques important for the math category of the board exam?

-Calculator techniques are important because they can significantly ease the process of solving complex mathematical problems during the exam, especially when the questions are in a multiple-choice format, which is common in board exams.

### What is the first topic covered in the video, and why is it considered basic?

-The first topic covered is algebra, specifically evaluating functions and simplifying expressions. It is considered basic because it forms the foundation for understanding more advanced mathematical concepts that will be discussed in subsequent videos.

### How does the video demonstrate the process of evaluating a function using a calculator?

-The video demonstrates the process by showing step-by-step instructions on how to input a given function into a calculator, such as 'f(x) = 2x^2 - 5x + 3', and then how to evaluate the function for specific values of x, like x = 2 and x = 4.

### What is the Remainder Theorem, and how is it applied in the video?

-The Remainder Theorem states that the remainder of a polynomial f(x) divided by (x - k) is equal to f(k). In the video, the theorem is applied to find the remainder when a given polynomial is divided by (x + 3) by evaluating the function at x = -3.

### How does the video use the Factor Theorem to determine if a given expression is a factor of a polynomial?

-The video uses the Factor Theorem by showing that if x - k is a factor of a polynomial, then the remainder when the polynomial is divided by x - k should be zero. This is demonstrated by evaluating the polynomial at different values of k and checking for a remainder of zero.

### What is the significance of assigning values to variables like x, y, and z when simplifying expressions using a calculator?

-Assigning values to variables is significant because it allows for the evaluation of the expression to determine if it simplifies to one of the given choices in a multiple-choice question. This trial-and-error method can help in identifying the correct simplification among the options provided.

### How can the video's approach to solving problem number four help in understanding the difference between f(x+1) and f(x)?

-The video's approach involves calculating the numerical values of f(x+1) and f(x) for a given x, and then finding the difference between these values. This process helps to understand how the function changes when the input is incremented by one, which is key to solving the problem.

### What is the process described in the video for evaluating a function with two variables, such as f(x, y) = 4x^3 + 3x^2y - 5xy^2 + y^3?

-The process involves inputting the function into the calculator, ensuring to use the alpha key to correctly input the variables and their respective powers. Then, the calculator is used to evaluate the function by inputting specific values for x and y, and the result is obtained by pressing the equal sign.

### How does the video guide viewers in simplifying complex rational expressions involving multiple variables?

-The video guides viewers by assigning values to the variables in the rational expression and using a calculator to evaluate the expression. It then compares the result with the given choices to find the one that matches the calculated value, indicating the correct simplification.

### Outlines

### ๐ Introduction to Engineering Calculator Techniques

This paragraph introduces the video's focus on calculator techniques essential for engineering board exam preparation. The speaker emphasizes the importance of these techniques for passing the math section of the exam, particularly for mechanical, electrical, and electronics engineering students. The tutorial is designed to cover various mathematical topics applicable to calculator use, starting with algebra and progressing to more complex subjects like derivatives, integrals, differential equations, and analytic geometry. The speaker clarifies that calculator techniques are only applicable to multiple-choice questions.

### ๐ข Demonstrating Function Evaluation Using a Calculator

The speaker provides a step-by-step guide on how to use a calculator to evaluate functions, using the function f(x) = 2x^2 - 5x + 3 as an example. They explain the process of turning on the calculator, inputting the equation, and evaluating it for specific values of x (in this case, x = 2 and x = 4). The explanation includes details about using the 'alpha' function to denote outputs and the importance of using a calculator that is allowed in the exam setting, such as the fx-570ES PLUS mentioned.

### ๐ Applying the Remainder Theorem with a Calculator

The paragraph explains how to use the Remainder Theorem to find the remainder of a polynomial function when divided by a linear factor. The speaker illustrates this with the function 2x^3 + 3x^2 - 2x + 25, showing how to set up the calculator to find the remainder when the function is divided by x + 3. The process involves equating x + 3 to zero to find the value of x for which f(x) will be the remainder, which in this case is x = -3, leading to a remainder of 4.

### ๐ Identifying Factors Using Calculator Techniques

The speaker discusses using calculator techniques to identify factors of the polynomial 3x^3 + 2x^2 - 32, given multiple-choice options. They explain the Factor Theorem, which states that if x - k is a factor of a polynomial, then f(k) should equal zero. The process involves inputting the polynomial into the calculator and evaluating it for each potential factor's value to determine if the remainder is zero, thus confirming it as a factor.

### ๐ Calculating Function Differences with a Calculator

This section describes how to calculate the difference between f(x+1) and f(x) for the function f(x) = 10^(x+1). The speaker suggests using a calculator to compute the numerical values of f(3) and f(2) and then subtracting the latter from the former to find the difference. The correct answer is identified through the process of elimination using multiple-choice options, with the correct answer being 900, corresponding to option D.

### ๐ Evaluating Multivariable Functions on a Calculator

The paragraph explains the process of evaluating a function with two variables, f(x, y) = 4x^3 + 3x^2y - 5xy^2 + y^3, using a calculator. The speaker demonstrates how to input the function into the calculator, assign values to x and y, and then evaluate the function for those values. The example given uses x = 1 and y = 2, resulting in an answer of -2, which corresponds to option C.

### ๐งฉ Simplifying Rational Expressions with a Calculator

The speaker presents a method for simplifying the expression (x^4 - y^4) / (x^4 - 2x^2y^2 + y^4) using a calculator. They suggest assigning values to x and y, inputting the expression into the calculator, and then evaluating it to find the simplified result. The example uses x = 2 and y = 3, leading to a simplified result that matches one of the multiple-choice options, identified as option B.

### ๐ข Simplifying Complex Exponential Expressions

This section covers the simplification of an expression involving exponential terms with variables x, y, and z. The speaker advises assigning values to these variables and inputting the expression into the calculator to find the simplified result. The example uses x = a, y = b, z = c with values a = 2, b = 3, and c = 4, resulting in a simplified expression that matches one of the multiple-choice options, identified as option A.

### ๐ Solving for Variables in Exponential Equations

The paragraph explains how to solve for x in an equation involving exponential terms with variable n. The speaker demonstrates the process of simplifying the equation by replacing 4^n with x and then solving for x using different values of n. The example shows that regardless of the value used for n, the result for x remains the same, indicating the solution for x in terms of n.

### ๐ Simplifying Expressions with Variables a and b

The speaker simplifies the expression a^(-2) * b^6 / (a^2 * b^(-1)) using a calculator with assigned values for a and b. The example uses a = 2 and b = 3, resulting in a simplified result of 2187/64. The speaker then compares this result with multiple-choice options to identify the correct answer, which is option C.

### ๐ Solving for Variable a in an Exponential Equation

The final paragraph demonstrates solving for the variable a in the equation a = 64^x * 4^y using a calculator. The speaker shows how to input the equation into the calculator, assign values to x and y, and compute the numerical value of a. The example uses x = 2 and y = 3, leading to a value for a that matches one of the multiple-choice options, identified as option D.

### Mindmap

### Keywords

### ๐กCalculator Techniques

### ๐กEngineering Board Exam

### ๐กAlgebra

### ๐กEvaluating Functions

### ๐กSimplifying Expressions

### ๐กMultiple Choice Questions

### ๐กRemainder Theorem

### ๐กFactor Theorem

### ๐กRational Expressions

### ๐กVariable Assignment

### ๐กSolving for a Variable

### Highlights

Introduction to calculator techniques for engineering board exams.

Importance of calculator techniques in passing the math section of board exams.

Tutorial covers algebra, evaluating functions, and simplifying expressions.

Explanation of using a calculator for evaluating functions with given values.

Demonstration of inputting equations into a calculator for evaluation.

Using the Remainder Theorem to find the remainder of a polynomial division.

Method to determine if a given expression is a factor of a polynomial.

How to use calculator techniques for multiple-choice questions effectively.

Process of evaluating functions with two variables using a calculator.

Simplifying complex algebraic expressions with the help of a calculator.

Assigning values to variables for the purpose of simplification in expressions.

Technique to solve for a variable in an equation using calculator input.

Storing calculated values in the calculator for future reference.

Solving equations involving exponential functions with variables.

Final problem walkthrough using calculator techniques for equation solving.

Conclusion and call to action to share the tutorial with others preparing for exams.

Emphasizing the practicality of mastering calculator techniques for exam success.