Problem 59 - Elementary and Intermediate Algebra

Question

Evaluate each expression. See Example 2 and \(3 .\) \(y^{4}-y^{3}+y^{2}+2 y-1\) for a. \(y=1\) b. \(y=-1\)

Step-by-Step Solution

Answer

For \( y = 1 \), the value is 2; for \( y = -1 \), the value is 0.

Step 1: Substitute y = 1 in the expression

Take the expression \[ y^4 - y^3 + y^2 + 2y - 1 \] and substitute \( y = 1 \): \[ (1)^4 - (1)^3 + (1)^2 + 2(1) - 1. \] Simplify each term: \( 1 - 1 + 1 + 2 - 1 = 2 \). Thus, the value of the expression for \( y = 1 \) is 2.

Step 2: Substitute y = -1 in the expression

Now take the same expression \[ y^4 - y^3 + y^2 + 2y - 1 \] and substitute \( y = -1 \): \[ (-1)^4 - (-1)^3 + (-1)^2 + 2(-1) - 1. \] Simplify each term: \( 1 + 1 + 1 - 2 - 1 = 0 \). Thus, the value of the expression for \( y = -1 \) is 0.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. Unlike equations, they do not have an "equals" sign. An algebraic expression can range from simple to complex based on the number and type of terms involved. For example,

• Monomial: a single term like \(7x\) or \(3y^2\)
• Binomial: two terms like \(4a - 5b\)
• Polynomial: multiple terms like \(x^3 - x^2 + x - 7\)

In the given exercise, the expression \(y^4 - y^3 + y^2 + 2y - 1\) is a polynomial with five terms. Understanding the structure of these expressions is crucial because it allows you to manipulate and solve problems involving them.

Mastering the Substitution Method

The substitution method is a technique used to evaluate algebraic expressions by replacing variables with specific values. This method is particularly useful when you need to calculate numerical outcomes based on given conditions.
To perform substitution:

• Identify the variable in the expression.
• Substitute the variable with the given value.
• Perform the arithmetic operations to simplify the expression to a single value.

In our exercise, we substituted \(y = 1\) and \(y = -1\) into the polynomial \(y^4 - y^3 + y^2 + 2y - 1\). Through this process, we directly replaced every occurrence of \(y\) with these numbers and calculated the final results.

Exploring Integer Powers of Variables

Integer powers of variables are expressions involving a variable raised to a whole number power. Each power indicates how many times you multiply the variable by itself. For instance, \(y^2\) means \(y \times y\), and \(y^3\) means \(y \times y \times y\).
These powers form crucial parts of polynomials and higher degree algebraic expressions. Key rules of integer powers include:

• \(a^0 = 1\) for any non-zero number \(a\)
• \(a^1 = a\)
• \(a^m \times a^n = a^{m+n}\)
• \(\frac{a^m}{a^n} = a^{m-n}\) when \(a eq 0\)

In our problem, powers like \(y^4\), \(y^3\), and \(y^2\) needed simplification after replacing \(y\) with specific values, emphasizing the importance of understanding how to handle powers in algebraic expressions.