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Chapter 4: Problem 54

Find each product. $$ (t-3)(t+8) $$

Short Answer

Expert verified
The product is \( t^2 + 5t - 24 \).

Step by step solution

01

Apply the Distributive Property

The distributive property states that for any three numbers a, b, and c, \[a(b + c) = ab + ac\]. In this case, you need to apply the distributive property twice.First, distribute \(t\) to both \(t\) and \(8\):\[(t-3)(t+8) = t(t+8) - 3(t+8)\]
02

Distribute Each Term

Now distribute each term within the parentheses:First distribute \(t\):\[t(t+8) = t^2 + 8t\]Then distribute \(-3\):\[-3(t+8) = -3t - 24\]
03

Combine Like Terms

Combine all the terms you acquired by distributing:\[t^2 + 8t - 3t - 24\]Combine the like terms \(8t\) and \(-3t\):\[t^2 + (8t - 3t) - 24 = t^2 + 5t - 24\]

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Key Concepts

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

distributive property
The distributive property is a fundamental concept in algebra and mathematics. It allows you to multiply a single term by each term inside a parenthesis. It is universally expressed as \(a(b + c) = ab + ac\). In the given problem, you use the distributive property twice:

1. Distribute \(t\) over both terms inside the parenthesis \((t+8)\):
\((t-3)(t+8) = t(t+8) - 3(t+8))\).

2. Now, distribute each term within the parenthesis:
\(t(t+8) = t^2 + 8t\) and \(-3(t+8) = -3t - 24\).

Distributing helps us to break down expressions into simpler parts, so that solving and simplifying becomes easier.
combining like terms
Combining like terms is essential for simplifying expressions. It involves adding or subtracting terms that have identical variable parts.

After applying the distributive property in the given exercise, you get \(t^2 + 8t - 3t - 24\).

1. Identify like terms: In this case, \(8t\) and \(-3t\) are like terms because they both have the variable \(t\).
2. Combine them: Add or subtract the coefficients of these like terms:
\(8t - 3t = 5t\).

This step simplifies the expression to \(t^2 + 5t - 24\), making it easier to understand and solve.
algebra
Algebra is the branch of mathematics that deals with symbols and the rules for manipulating those symbols. It encompasses everything from simple equations to complex functions. In this problem, algebraic principles such as the distributive property and combining like terms are used to simplify and solve polynomial expressions.

Understanding algebra involves learning several key operations and rules:
  • Distributive Property
  • Combining Like Terms
  • Solving Equations
  • Factoring

Practicing problems like the given exercise helps build a strong foundation in algebra. It trains you to look for patterns and use the appropriate rules to simplify and solve equations effectively.

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Most popular questions from this chapter

Find each product. $$ -5 r(r+1)^{3} $$

\begin{aligned} &\text { Find the difference of the sum of } 5 x^{2}+2 x-3 \text { and } x^{2}-8 x+2 \text { and the sum of } 7 x^{2}-3 x+6\\\ &\text { and }-x^{2}+4 x-6 \end{aligned}

The special product \((x+y)(x-y)=x^{2}-y^{2}\) can be used to perform some multiplications.Example: $$\begin{array}{l|l}51 \times 49 & 102 \times 98 \\\=(50+1)(50-1) & =(100+2)(100-2) \\\=50^{2}-1^{2} & =100^{2}-2^{2} \\\=2500-1 & =10,000-4 \\\=2499 & =9996\end{array}$$ Use this method to calculate each product mentally. $$ 201 \times 199 $$

Find each product. $$ (3 t-4 s)(t+3 s) $$

Perform each indicated operation. \(\left(-2 b^{6}+3 b^{4}-b^{2}\right)+\left(b^{6}+2 b^{4}+2 b^{2}\right)\)
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