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Chapter 1: Problem 19

Find the value for \((a) x=4\) and \((b) x=6\). \(3 x^{2}+x\)

Short Answer

Expert verified
For (a) x = 4, the value is 52. For (b) x = 6, the value is 114.

Step by step solution

01

- Understand the equation

Identify the given equation: \[ 3x^2 + x \]
02

- Substitute value for (a) x = 4

Substitute \( x = 4 \) into the equation. \[ 3(4)^2 + 4 \]
03

- Simplify for (a) x = 4

Calculate the value: \[ 3(16) + 4 = 48 + 4 = 52 \]
04

- Substitute value for (b) x = 6

Substitute \( x = 6 \) into the equation. \[ 3(6)^2 + 6 \]
05

- Simplify for (b) x = 6

Calculate the value: \[ 3(36) + 6 = 108 + 6 = 114 \]

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Key Concepts

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

substitution method
The substitution method is a straightforward way to evaluate polynomials. It involves replacing the variable in a polynomial with a given value. In our exercise, we replaced \( x \) with specific numbers (4 and 6) to find the resulting value of the polynomial. This process requires careful attention to substitution to avoid errors. For example, for \( x = 4 \), we substituted it in \( 3x^2 + x \) to get \( 3(4)^2 + 4 \). It's a primary method for evaluating expressions and understanding how changing the variable affects the polynomial's outcome.
Steps:
  • Identify the polynomial equation.
  • Replace the variable with the given number.
  • Follow order of operations (PEMDAS/BODMAS) to simplify.
simplification
Simplification is the process of breaking down a complex expression into a more manageable form. After substituting the variable, the next step is simplifying the expression. Proper simplification follows the order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). For instance, in our example when \( x = 4 \), we simplified \( 3(4)^2 + 4 \) to \( 3(16) + 4 = 48 + 4 = 52\). This step is essential to ensure accurate results.
Key points:
  • Always follow the correct order of operations.
  • Simplify step-by-step without skipping processes.
  • Double-check calculations to avoid errors.
polynomials
Polynomials are algebraic expressions that consist of variables and coefficients, with only non-negative integer exponents of the variables. Examples include \( 3x^2 + x \), which was used in our exercise. Understanding polynomials involve recognizing their structure and knowing how to manipulate them. Key components include terms (e.g., \(3x^2 \) and \( x \) are terms), coefficients (the numbers multiplied by the variables, like 3 and 1), and exponents (which show the power to which the variable is raised). Grasping these basics is crucial for solving and simplifying polynomial expressions.
Characteristics:
  • They can have one or more terms.
  • Terms are separated by + or - signs.
  • Polynomials are classified by degree, based on the highest power of the variable.

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