Problem 32 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ 5 a^{2}-6 a+11, \text { if } a=0 $$

Step-by-Step Solution

Answer

The value is 11.

Step 1: Substitute Value of a

The expression to solve is \(5a^2 - 6a + 11\). We need to find the value when \(a = 0\). Substitute \(0\) for \(a\) in the expression. This gives \(5(0)^2 - 6(0) + 11\).

Step 2: Simplify Powers and Products

First, simplify the power and product terms in the expression. \(5(0)^2\) simplifies to \(5 imes 0\), which equals \(0\). Also, \(-6(0)\) simplifies to \(-6 imes 0\), which is also \(0\). This reduces the expression to \(0 - 0 + 11\).

Step 3: Combine the Results

Since both the products resulted in zero, combine the remaining terms: \(0 - 0 + 11 = 11\).

Key Concepts

Polynomial Evaluation

Polynomial evaluation is the process of finding the value of a polynomial expression when the variable is replaced with a specific number. Let's break down what happens when evaluating a polynomial:

• The polynomial is an expression consisting of terms that involve variables raised to various powers.
• To evaluate the polynomial, each variable in the expression is substituted with a given number.

Consider the polynomial expression from the exercise: \[5a^2 - 6a + 11\]When we substitute the value of \(a = 0\), the aim is to compute the value of the expression so that the variable \(a\) is completely replaced, and the only tasks left are arithmetic calculations. This involves:- Substituting the value of \(a\) into the polynomial.- Simplifying the expression by carrying out basic arithmetic operations.For example, substituting \(a = 0\) yields \(5(0)^2 - 6(0) + 11\), which simplifies as we calculate powers and products. In this case, the simplified result is \(11\).

Algebraic Expressions

Algebraic expressions are combinations of letters and numbers linked by operations like addition, subtraction, multiplication, or division. These are foundational in algebra and help in representing general mathematical statements.

• They contain terms which can be constants (like numbers) or coefficients multiplied by variables.
• Algebraic expressions can include one or more terms.

In the expression \(5a^2 - 6a + 11\), here:- \(5a^2\) is a term where \(5\) is the coefficient, and \(a^2\) is the variable term.- \(-6a\) has \(-6\) as the coefficient, and \(a\) is the variable.- \(11\) is a constant term, which means it doesn’t change when the value of \(a\) changes.Every term in an algebraic expression plays a role, whether it is a constant, a coefficient, or a variable. Understanding these roles helps in simplifying and evaluating expressions efficiently.

Substitution Method

The substitution method is a fundamental technique in algebra used to replace variables in expressions with numerical values. This allows us to calculate a numerical value for an expression that otherwise contains indefinable variables.How does substitution work? Let’s look at the expression from the exercise:- The expression \(5a^2 - 6a + 11\) doesn't provide any value until \(a\) is given a numerical value.Using substitution:1. **Identify the variable**: In this case, \(a\) is the variable.2. **Assign a given value**: Replace \(a\) with \(0\).3. **Calculate the result**: Perform arithmetic operations to simplify the expression while following order of operations, PEMDAS/BODMAS.The substitution method streamlines the process of solving expressions and equations and is a valuable tool for evaluating polynomials like the one in our exercise.