Problem 7 - Elementary and Intermediate Algebra

Question

To ____ the polynomial \(x^{2}-2 x+1\) for \(x=6,\) we substitute 6 for \(x\) and follow the rules for the order of operations.

Step-by-Step Solution

Answer

The polynomial evaluates to 25 when \(x = 6\).

Step 1: Substitute 6 into the polynomial

Replace every instance of the variable \(x\) in the polynomial \(x^2 - 2x + 1\) with 6. This gives us the expression \(6^2 - 2 \times 6 + 1\).

Step 2: Calculate the exponent

Evaluate \(6^2\), which means calculating \(6\times6\). This results in 36. So, the expression becomes \(36 - 2 \times 6 + 1\).

Step 3: Multiply

Next, we need to multiply \(-2\) by 6. This results in \(-12\). Now the expression is \(36 - 12 + 1\).

Step 4: Perform addition and subtraction

Now, perform the subtraction and then the addition from left to right: \(36 - 12 = 24\), and then \(24 + 1 = 25\).

Key Concepts

Substitution Method

The substitution method is a handy technique for evaluating polynomial expressions. It involves replacing the variable in the expression with a given number. In our exercise, we are given the polynomial \(x^{2} - 2x + 1\) and need to evaluate it for \(x = 6\). By substituting, we replace every occurrence of \(x\) with the number 6. This transforms our polynomial into numerical form, making it simpler to calculate. Using the substitution method allows us to focus on numerical computation rather than manipulation of algebraic expressions. It's a straightforward process:

• Identify the variable to substitute.
• Replace the variable with the given number in the expression.

Once substituted, we then move on to evaluating the resulting expression by following the order of operations.

Order of Operations

After substitution, the key to correctly evaluating the expression \(6^2 - 2 \times 6 + 1\) is applying the order of operations. The order of operations can be remembered by the acronym PEMDAS:

• P: Parentheses
• E: Exponents
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

It's important to follow these steps in sequence to ensure accurate results. For our example, we start with exponents. Calculating the exponent first: evaluate \(6^2\), resulting in 36. Next, perform multiplication first, as dictated by PEMDAS, we handle the multiplication: \(-2 \times 6\), equating to \(-12\). Lastly, we handle addition and subtraction in order from left to right, ending with an evaluated expression of 25. Using the order of operations ensures that we accurately simplify complex expressions.

Polynomial Expressions

Polynomial expressions consist of variables and coefficients combined using addition, subtraction, and multiplication, raised to non-negative integer powers. In the polynomial \(x^2 - 2x + 1\), we see:

• The term \(x^2\) with a power of 2.
• The linear term \(-2x\) with a power of 1.
• The constant term 1.

Each part plays a role in forming the full expression. Polynomials can be evaluated by substituting a value into the variable, simplifying using arithmetic operations, and applying the order of operations.Understanding how polynomial expressions are structured helps in breaking down each part for evaluation. Recognizing terms, coefficients, and the highest degree can guide you through simplifications efficiently. Whether it's algebraic manipulation or substitution, having a clear grasp on the components of polynomials streamlines working through mathematical problems with ease.