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Chapter 2: Problem 72

Evaluate the expression. $$t^{2}-7 t+12 \text { when } t=8$$

Short Answer

Expert verified
The final evaluation of the expression when \(t=8\) is 20.

Step by step solution

01

Substitution

Replace \(t\) with 8 in the expression. This gives: \(8^{2} - 7*8 + 12\).
02

Follow Operational Order

Perform the operations in the correct order. Start with the exponent, then perform multiplication and at last addition. This results in: \(64 - 56 + 12\).
03

Perform Addition/Subtraction

Continue performing operations, doing addition and subtraction from left to right. This results in: \((64 - 56) + 12 = 8 + 12 = 20\).

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Key Concepts

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

Substitution
In mathematics, substitution is a straightforward concept. You simply replace a variable in an expression with a given value. In this exercise, the variable is \( t \), and we substitute it with the number 8. This transforms the original expression \( t^2 - 7t + 12 \) into \( 8^2 - 7 \times 8 + 12 \).

Substitution is like putting a specific number into a recipe where a variable is used. It's important to make sure every occurrence of the variable is replaced with the number you are given.

For this concept, make sure to double-check the transformation of each term to ensure accuracy. A small mistake can lead to an incorrect solution.
Order of Operations
The order of operations is a critical rule in mathematics. It tells us which calculations to perform first in an expression to ensure everyone gets the same answer. In this exercise, after substituting \( t = 8 \), the expression becomes \( 8^2 - 7 \times 8 + 12 \).

The order to follow is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)). Here鈥檚 the process for our expression:
  • Exponents: Calculate \( 8^2 \) which is 64.
  • Multiplication: Compute \( 7 \times 8 \), resulting in 56.
  • Addition/Subtraction: Perform from left to right: \( 64 - 56 + 12 \).
This correct order ensures accuracy and consistency.
Polynomial Evaluation
Evaluating a polynomial is the process of calculating the output of the polynomial given a particular input for its variable. The exercise in question involves a polynomial expression \( t^2 - 7t + 12 \) with \( t = 8 \).

Once the value (8) is substituted into the expression, following the order of operations as described allows us to efficiently simplify and compute:
  • The expression converts to \( 8^2 - 7 \times 8 + 12 \).
  • Using the order of operations: \( 64 - 56 + 12 \).
  • This breaks down further into \( 8 + 12 = 20 \).
Polynomial evaluation involves careful substitution and following of operations steps. Once mastered, it becomes a valuable skill in algebra and beyond.

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Most popular questions from this chapter

INVESTING MONEY You receive \(\$ 5000\). You decide to invest the money in a one-year bond paying \(2 \%\) interest and in a one-year certificate of deposit paying \(6 \%\) interest. \- Let \(m\) represent the amount of money invested in the one-year bond. Write a function that represents the total amount of money \(T\) that you have after one year. Simplify the function.

RECIPROCALS Find the reciprocal. $$ 7 $$

SIMPLIFYING EXPRESSIONS Simplify the expression by combining like terms. $$ \frac{7}{9} w+\left(-\frac{2}{3}\right) w $$

EVALUATING EXPRESSIONS Evaluate the expression. $$ -7+9-8 $$

DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the expression without parentheses. $$ x(x+1) $$
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