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Chapter 0: Problem 50

Perform the indicated operations and simplify. $$ \left(2 a^{2}+11 b^{3}\right)^{2} $$

Short Answer

Expert verified
The simplified form is \(4a^{4} + 44a^{2}b^{3} + 121b^{6}\).

Step by step solution

01

Expand the Expression

The expression \(\big(2a^{2} + 11b^{3}\big)^{2}\) can be expanded using the binomial theorem \((x + y)^{2} = x^{2} + 2xy + y^{2}\).
02

Identify Components

In the given expression, \(x = 2a^{2}\) and \(y = 11b^{3}\).
03

Square Each Term

Calculate \(x^{2}\) and \(y^{2}\): \(x^{2} = \big(2a^{2}\big)^{2} = 4a^{4}\) and \(y^{2} = \big(11b^{3}\big)^{2} = 121b^{6}\).
04

Calculate the Product of Terms and Double It

Calculate \((2xy)\): \(2 \times 2a^{2} \times 11b^{3} = 44a^{2}b^{3}\).
05

Combine the Results

Put all terms together: \(4a^{4} + 44a^{2}b^{3} + 121b^{6}\).

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Key Concepts

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

Polynomial Operations
Polynomial operations involve addition, subtraction, multiplication, and division of polynomials. In this exercise, we're focusing on multiplication through the binomial expansion. When working with polynomials:
鈥� Pay attention to the coefficients and the variables.
鈥� Follow the algebraic rules for combining like terms.
In the expression \(\big(2a^{2} + 11b^{3}\big)^{2}\), we treat it as a binomial where each term is part of a polynomial expression. We need to expand it using polynomial operations, adhering to the binomial theorem.
Remember that polynomials are expressions with multiple terms. Properly executing polynomial operations ensures we get the correct expanded form.
Exponents in Algebra
Exponents in algebra signify how many times a number (the base) is multiplied by itself. For example, in \((2a^{2})^{2}\), the number \(2a^{2}\) is raised to the power of 2, meaning it is multiplied by itself. Key rules to remember:
鈥� \( (a^m)^n = a^{mn} \) means multiplying the exponents.
鈥� When multiplying like bases, add the exponents: \( a^m \times a^n = a^{m+n} \.\)
The binomial theorem is rooted in these exponent rules. In Step 3, we squared each term in the polynomial:
鈥� \((2a^{2})^{2} = 4a^{4}\)
鈥� \((11b^{3})^{2} = 121b^{6}\)
This demonstrates applying exponent rules to simplify and perform polynomial operations accurately.
Binomial Expansion
The binomial theorem is a fundamental theorem in algebra used to expand expressions raised to a power. For a binomial \( (x + y)^2\), the expansion is:
\( x^2 + 2xy + y^2 \).
This serves as a template to broaden more complex expressions. Our given problem \( (2a^{2} + 11b^{3})^{2}\) specifies:
鈥� \( x = 2a^{2} \)
鈥� \( y = 11b^{3} \)
Following the theorem, we:
  • Square each part:
    • -- \((2a^{2})^{2} = 4a^{4}\)-- \((11b^{3})^{2} = 121b^{6}\)
  • Double the product of the terms:
    • -- \( 2 \times 2a^{2} \times 11b^{3} = 44a^{2}b^{3}\)
Lastly, combine all the terms to get the simplified expanded form:
\( 4a^{4} + 44a^{2}b^{3} + 121b^{6} \).
This is the result of the binomial expansion technique.

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Most popular questions from this chapter

The total national expenditure for health care has been increasing since the year 2000 . For privately insured individuals in the United States, the following models give the total amount spent for health insurance premiums \(I\) (in \$ billions) and the total amount spent on other out-of-pocket health- related expenses \(P\) (in \$ billions). (Source: U.S. Centers for Medicare \& Medicaid Services, www.census.gov) \(I=45.58 x+460.1 \quad\) Total spent on health insurance premiums \(x\) years since 2000. \(P=10.86 x+191.5 \quad\) Other out-of-pocket health-related expenses \(x\) years since 2000. Determine the total expenditure for other health-related out-of-pocket expenses for the year \(2005 .\)

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If \(n>0,\) then \(-|n|=\) _____ .

Write the polynomial in descending order. Then identify the leading coefficient and degree of the polynomial. $$ \frac{4}{5} c^{2}+c^{5} $$
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