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Chapter 4: Problem 23

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

Short Answer

Expert verified
The product is \(9t^3 - 6t^2 + t\).

Step by step solution

01

- Recognize the binomial expansion formula

Use the binomial expansion formula ewline \((a - b)^2 = a^2 - 2ab + b^2\).Identify \(a = 3t\) and \(b = 1\).
02

- Apply the binomial expansion

Expand \((3t - 1)^2\) using the binomial formula:ewline \((3t - 1)^2 = (3t)^2 - 2 \cdot (3t) \cdot 1 + 1^2\)ewline Calculate each term: ewline \((3t)^2 = 9t^2\), ewline \(-2 \cdot (3t) \cdot 1 = -6t\),ewline \(1^2 = 1\).
03

- Combine terms in expansion

Combine the terms from the expansion:ewline \((3t - 1)^2 = 9t^2 - 6t + 1\).
04

- Distribute \(t\) over the expanded result

Now multiply \(t\) with each term of the expansion:ewline \[ t(9t^2 - 6t + 1) = t \cdot 9t^2 - t \cdot 6t + t \cdot 1 \] ewline Calculate each product: ewline \[ t \cdot 9t^2 = 9t^3, ewline - t \cdot 6t = -6t^2, ewline t \cdot 1 = t \].
05

- Combine the final terms

Combine all terms to get the final polynomial:ewline \[ 9t^3 - 6t^2 + t \].

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Key Concepts

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

Binomial Expansion
The binomial expansion is a method used to expand expressions that are raised to a power, typically in the form ewline\((a \text{ or } b)^n\). Here, we were given ewline\((3t - 1)^2\). ewlineUsing the expansion formula ewline\((a - b)^2 = a^2 - 2ab + b^2\), ewlinewe identified the terms ewline\(a\) and ewline\(b\) in our expression: ewline\(a = 3t\) and ewline\(b = 1\). We then apply the formula substituting the values of ewline\(a\) and ewline\(b\): ewline\((3t - 1)^2 = (3t)^2 - 2 \cdot 3t \cdot 1 + 1^2\). By calculating each part:
  • (3t)^2 = 9t^2
  • -2 \cdot (3t) \cdot 1 = -6t
  • 1^2 = 1
we complete the expansion to get ewlne\(9t^2 - 6t + 1\).
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In our exercise, ewline\((3t - 1)^2\) and ewline\(t(3t-1)^2\) are examples of such expressions.ewlineUnderstanding factors, terms, and coefficients is key:
  • Factors are quantities being multiplied, for instance, in ewline\(3t\) 3 and ewline\(t\) are factors.
  • Terms are parts of the expression separated by ewline\(+\text{ or }-\), for example, ewline\(9t^2, -6t,\text{ and }1\).
  • Coefficients are numerical factors of terms, like the ewline\(9, -6,\text{ and }1\) in our expanded polynomial.
ewlineCombining the terms carefully is essential when manipulating such expressions.
Distributive Property
The distributive property is a key rule in algebra used to multiply a single term by each term within a set of parentheses.ewlineWhen we have ewline\(t(9t^2 - 6t + 1)\), we apply this property:ewlineMultiply ewline\(t\) by each part:
  • \(t \codot 9t^2 = 9t^3\)
  • \(-t \codot 6t = -6t^2\)
  • \(t \codot 1 = t\)
The property ensures consistent and accurate expansion of terms. Combining the distributed products (or terms) correctly is important for solving the expression carefully.
Polynomial Addition
Polynomial addition involves combining like terms to form a simplified polynomial. After distributing ewline\(t\), we end up with
  • \(9t^3\)
  • \(-6t^2\)
  • \(t\)
We then combine these terms to arrive at the final answer:ewline\(9t^3 - 6t^2 + t\). ewlineEach step in combining terms should be carried out methodically to ensure no mistakes, as any error can lead to incorrect expressions.

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Most popular questions from this chapter

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (5 k+3 q)^{2} $$

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (x+7)^{2} $$

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. $$ 101 \times 99 $$

Find each product. Recall that \(a^{2}=a \cdot a\) and \(a^{3}=a^{2} \cdot a\). $$ (3 m-1)^{2} $$

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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