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Chapter 4: Problem 17

Find each product. $$ 3 t(4 t+1)^{2} $$

Short Answer

Expert verified
48t^3 + 24t^2 + 3t

Step by step solution

01

Expand the Square

First, expand the expression inside the parentheses \[ (4t + 1)^2 \].Using the formula \[ (a + b)^2 = a^2 + 2ab + b^2 \], where \(a = 4t\) and \(b = 1\), we get:\[ (4t)^2 + 2(4t)(1) + 1^2 = 16t^2 + 8t + 1 \].
02

Distribute the Outside Term

Distribute the term \(3t\) through the expanded polynomial: \[ 3t(16t^2 + 8t + 1) \].
03

Multiply Each Term

Multiply \(3t\) by each term inside the parentheses:\[ 3t \times 16t^2 = 48t^3 \]\[ 3t \times 8t = 24t^2 \]\[ 3t \times 1 = 3t \].Finally, add the terms together to get the product:\[ 48t^3 + 24t^2 + 3t \].

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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 key principle in algebra that helps simplify expressions and solve equations. This property states that multiplying a single term by each term inside a set of parentheses is equivalent to distributing the multiplication across all terms: \[ a(b + c) = ab + ac \]. In our problem, we applied the distributive property by multiplying \(3t\) with each term in the expanded polynomial \(16t^2 + 8t + 1\). Here鈥檚 how it works:
  • First, \(3t\) is multiplied by \(16t^2\) to get \(48t^3\).
  • Second, \(3t\) is multiplied by \(8t\) to get \(24t^2\).
  • Finally, \(3t\) is multiplied by \(1\) to get \(3t\).
The result is a simplified expression: \(48t^3 + 24t^2 + 3t\). This demonstrates how the distributive property makes it easier to expand and simplify expressions, leading to more manageable results.
Expanding Polynomials
Expanding polynomials involves distributing multiplication over addition or subtraction within the polynomial. Essentially, we rewrite a polynomial in an extended form.
In our exercise, we had to expand \((4t + 1)^2\). Using the binomial expansion formula \((a + b)^2 = a^2 + 2ab + b^2\) with \(a = 4t\) and \(b = 1\), we achieved the expansion by:
  • Calculating \( (4t)^2 \) to get \( 16t^2 \).
  • Calculating \( 2(4t)(1) \) to get \( 8t \).
  • Calculating \( 1^2 \) to get \( 1 \).
Combining these terms gives us \( 16t^2 + 8t + 1 \). Expanding polynomials in this manner helps us in further calculations, like distributing multiplication over the expanded terms.
Understanding Algebraic Expressions
An algebraic expression is a combination of variables, constants, and arithmetic operators. It's a fundamental concept in algebra used to model real-world scenarios and solve various mathematical problems.In our exercise, the expression \(3t(4t + 1)^2\) is an example of a more complex algebraic expression. Here鈥檚 a broader understanding:
  • Variables: Letters like \(t\) that represent unknown values.
  • Constants: Fixed numbers like \(3\) and \(1\).
  • Operators: Symbols like \(+\) and \(\times\) used for arithmetic operations.
To solve the expression, we first expanded the polynomial within the parentheses and then used the distributive property. Combining these concepts, we simplified the expression to a final form \(48t^3 + 24t^2 + 3t\). Understanding algebraic expressions and their components helps in breaking down and solving mathematical problems efficiently.

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Most popular questions from this chapter

Use scientific notation to calculate the answer to each problem. Write answers in scientific notation. $$ \frac{0.000000081(0.000036)}{0.00000048} $$

Find each product. Does \((a+b)^{2}\) equal \(a^{2}+b^{2}\) in general? Explain.

The special product $$ (x+y)(x-y)=x^{2}-y^{2} $$ can be used to perform some multiplication problems. Here are two examples. $$ \begin{aligned} 51 \times 49 &=(50+1)(50-1) \\ &=50^{2}-1^{2} \\ &=2500-1 \\ &=2499 \end{aligned} \quad | \begin{aligned} 102 \times 98 &=(100+2)(100-2) \\ &=100^{2}-2^{2} \\ &=10,000-4 \\ &=9996 \end{aligned} $$ Once these patterns are recognized, multiplications of this type can be done mentally. Use this method to calculate each product mentally. $$ 103 \times 97 $$

Find each product. $$ q(5 q-1)(5 q+1) $$

Each statement contains a number in boldface italics. Write the number in scientific notation. In 2007 , assets of the insured commercial banks in the United States totaled about 13,039,000,000,000 dollars. (Source: U.S. Federal Deposit Insurance Corporation.)
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