Problem 49 - Fundamentals of Mathematics

Question

Find the value of each expression. $$-8 y^{2}+6 y+11, \text { if } y=0$$

Step-by-Step Solution

Answer

The value of the expression is 11 when \(y = 0\).

Step 1: Substitute y with 0

Given the expression \(-8y^{2} + 6y + 11\). First, substitute \(y = 0\) into this expression.

Step 2: Simplify the expression

After substitution, the expression becomes:\(-8(0)^{2} + 6(0) + 11\). Simplify each term separately.

Step 3: Evaluate each term

1. Calculate \(-8(0)^{2} = 0\) because any number multiplied by zero is zero. 2. Calculate \(6(0) = 0\) for the same reason.3. The constant \(11\) remains as it is.

Step 4: Add terms together

Now, add all the simplified terms together:\(0 + 0 + 11 = 11\). This is the final result of the expression when \(y = 0\).

Key Concepts

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, where the variables can be raised to whole number exponents. These terms are connected either by addition or subtraction. In simpler terms, a polynomial might look something like this:

• The expression \(3x^2 + 5x - 7\) is a polynomial.
• Each part separated by a plus or minus sign is known as a term.

The expression \(-8y^2 + 6y + 11\)\, used in the exercise, is a polynomial in the variable \(y\). Here, \(-8y^2\) is a quadratic term, \(6y\) is a linear term, and \(11\) is a constant term. Such expressions are common in algebra, and understanding them forms the basis for more advanced topics. When working with polynomials, it's vital to identify each term by its degree, which is the highest exponent of the variable in that term. This is crucial for properly managing and simplifying expressions.

Substitution Method

The substitution method is a key technique in algebra that allows us to replace a variable in an equation or expression with a specific value. This process helps to evaluate the expression at particular values of the variable. For example, considering the expression \(-8y^2 + 6y + 11\), we substitute \(y\) with \(0\) to find the expression's value when \(y = 0\). This involves:

• Replacing all instances of \(y\) with \(0\).
• Calculating each term separately using the substituted value.

So, the expression changes as follows: \(-8(0)^2 + 6(0) + 11\). The substitution method is essential when solving equations or evaluating expressions to determine specific values.

Simplifying Expressions

Simplifying expressions involves reducing an expression to its most basic form where no further calculations or reductions are possible. This often means carrying out arithmetic operations and combining like terms. After substituting in the expression \(-8(0)^2 + 6(0) + 11\), we simplify it:

• First, calculate \(-8(0)^2 = 0\) since any number multiplied by zero is zero.
• Then calculate \(6(0) = 0\) using the same logic.
• Finally, keep the constant \(11\) as it is unaffected by any operations.

After simplifying each component, the expression becomes \(0 + 0 + 11\) which equals \(11\). This process of simplification not only makes the expression easier to understand but also provides the final answer when evaluating expressions using substitution.