Problem 100 Evaluate each expression for \(x... [FREE SOLUTION]

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Chapter 4: Problem 100

Evaluate each expression for $x=3 .$ See Sections 1.3 and 1.6. $$ 3 x^{2}-3 x+4 $$

Short Answer

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Step by step solution

Substitute the value of x

Replace the variable $ x $ with 3 in the expression $ 3x^2 - 3x + 4 $. So it becomes $ 3(3)^2 - 3(3) + 4 $.

Calculate the exponent

Evaluate $ 3^2 $ which is equal to 9. The expression now becomes $ 3(9) - 3(3) + 4 $.

Multiply

Multiply the constants by the results of the exponents: $ 3 \times 9 = 27 $ and $ 3 \times 3 = 9 $. The expression now is $ 27 - 9 + 4 $.

Perform addition and subtraction

First, subtract $ 27 - 9 = 18 $, then add $ 18 + 4 = 22 $. The final value of the expression is 22.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution

One of the initial steps in evaluating an algebraic expression is **substitution**. This means replacing a variable in the expression with a given number. Here, the variable is 'x' and we substitute it with 3. So, when we take the expression $3x^2 - 3x + 4$ and plug in 3 for every 'x', it becomes $3(3)^2 - 3(3) + 4$. Substitution simplifies the expression, making it easier to evaluate.

Order of Operations

To solve any algebraic expression correctly, it's crucial to follow the **order of operations**. The order is also known by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). For our expression, $3(3)^2 - 3(3) + 4$, we first calculate the **exponent**: $3^2$ which is 9. Next, we follow multiplication: $3 \times 9$ and $3 \times 3$. Only after handling both exponents and multiplications do we move on to addition and subtraction. This ensures the accuracy of the result.

Polynomial Expressions

A **polynomial expression** is an algebraic expression with multiple terms. In this case, we are working with $3x^2 - 3x + 4$. Polynomials can have various degrees. The degree is determined by the highest exponent. Here, the highest exponent is 2 (in the term $x^2$), so this is a second-degree polynomial. Understanding polynomials' structure helps in identifying and organizing the operations needed for evaluation.

Arithmetic Operations

The final step involves basic **arithmetic operations**: addition, subtraction, multiplication, and division. After substituting and simplifying the polynomial, we have three operations left: **multiplying**, **subtracting**, and **adding**. In our example, $3(3)^2 - 3(3) + 4$ becomes $27 - 9 + 4$ after performing the multiplications. From there, we subtract: $27 - 9 = 18$, and then add: $18 + 4 = 22$. Attention to each arithmetic operation ensures we correctly evaluate the expression.

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91影视

Evaluate each expression for \(x=3 .\) See Sections 1.3 and 1.6. $$ 3 x^{2}-3 x+4 $$

Short Answer

Step by step solution

Substitute the value of x

Calculate the exponent

Multiply

Perform addition and subtraction

Key Concepts

Substitution

Order of Operations

Polynomial Expressions

Arithmetic Operations

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