Problem 31 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ 4 y^{2}+3 y+1, \text { if } y=-2 $$

Step-by-Step Solution

Answer

The value of the expression is 11.

Step 1: Substitute the Given Value

First, substitute the given value, \( y = -2 \), into the expression \( 4y^2 + 3y + 1 \). This gives us: \( 4(-2)^2 + 3(-2) + 1 \).

Step 2: Calculate the Squared Term

Calculate \((-2)^2\), which is \(4\). Substitute this back into the equation to get: \( 4(4) + 3(-2) + 1 \).

Step 3: Multiply the Coefficients

Now multiply the terms: \(4 \times 4\) gives \(16\), and \(3 \times (-2)\) gives \(-6\). The equation now is: \(16 - 6 + 1\).

Step 4: Solve the Simplified Expression

Perform the arithmetic to solve the expression: \(16 - 6\) equals \(10\), and adding \(1\) gives \(11\).

Key Concepts

Substitution Method

The substitution method is a straightforward approach for evaluating algebraic expressions. The core idea is to replace the variable with a given value, simplifying the computation.
To use the substitution method, follow these steps:

• Identify the variable in the expression and note down its given value.
• Replace every occurrence of the variable with the given value in the expression.
• Compute each part of the expression based on arithmetic operations like addition, subtraction, multiplication, and division.

The goal is to find the value of the expression by resolving all arithmetic operations after the substitution. For instance, in the exercise, the expression is altered by replacing the variable \(y\) with \(-2\), transforming it to \(4(-2)^2 + 3(-2) + 1\). Following this, the expression is solved step by step.
This method is particularly useful when dealing with variables whose values are known, allowing us to ascertain the expression's value efficiently.

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, involving only operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
In the given problem, the expression \(4y^2 + 3y + 1\) is a polynomial. Here's why:

• It includes terms such as \(4y^2\), \(3y\), and a constant term, \(1\).
• The highest exponent of the variable \(y\) in the polynomial is \(2\), making it a quadratic polynomial.
• Each term is separated by addition or subtraction, making the entire expression a sum of multiple terms.
  

Working with polynomials involves handling terms with different powers of variables. Understanding polynomials are vital in algebra as they form the basis for more complex mathematical concepts. They are utilized in equations and functions, analyzing curves, and are integral parts of calculus.

Arithmetic Operations

Arithmetic operations form the backbone of algebraic manipulations. These operations include the basic operations like addition, subtraction, multiplication, and division.
Here's how they play a role in evaluating polynomials:

• Addition: Combine coefficients or constants when they are being added. In our expression, we perform addition in steps like adding the constant \(1\) at the end.
• Subtraction: Deduct one term from another, such as reducing the value by subtracting \(-6\) in the expanded expression.
• Multiplication: Similar terms are multiplied, typically done through distributing coefficients, like \(4 \times 4 = 16\) or \(3 \times (-2)\).
• Exponentiation: Squaring or raising to any power, as seen with \(-2\)^2 = 4.

Step-by-step application of these operations is crucial for simplifying expressions and solving equations. As we execute these operations methodically, the final simplified form or value of the expression emerges, leading us to the solution. For instance, combining \(16\), subtracting \(6\), and adding \(1\) ultimately leads to the final answer \(11\). This sequential process exemplifies precision and careful execution of arithmetic operations.