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Chapter 5: Problem 78

Perform the operations and simplify. $$\left(5 a^{4}-2 b\right)^{2}$$

Short Answer

Expert verified
25a^8 - 20a^4b + 4b^2

Step by step solution

01

- Write Down Expression

First, write the expression to be simplified: \[ (5a^4 - 2b)^2 \]
02

- Apply the Binomial Theorem

The binomial theorem states that \[ (x - y)^2 = x^2 - 2xy + y^2 \] Using this theorem, identify that \[ x = 5a^4 \] and \[ y = 2b \]
03

- Square Each Term

Square each part of the binomial individually: \[ (5a^4)^2 = 25a^8 \] \[ (2b)^2 = 4b^2 \]
04

- Calculate the Middle Term

The middle term is calculated as \[ -2xy \] Simplify this as follows: \[ -2(5a^4)(2b) = -20a^4b \]
05

- Combine All Terms

Now combine all the terms: \[ (5a^4 - 2b)^2 = 25a^8 - 20a^4b + 4b^2 \]

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Key Concepts

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

Simplification
Simplification is the process of making an expression simpler or more compact without changing its value. This often involves removing parentheses, combining like terms, and performing arithmetic operations.
We simplify expressions to make them easier to understand or solve. In our exercise, we take a complex expression and break it down using the binomial theorem to get a more straightforward polynomial form.
Here鈥檚 how simplification works in this context:
  • We start with \((5a^4 - 2b)^2\).
  • Apply the binomial theorem formula to expand the expression pass-by-pass.
  • Calculate each term individually and then combine them.
As you practice more, you'll get better at spotting opportunities to simplify expressions easily.
Algebraic Expressions
Algebraic expressions are mathematical phrases that include numbers, variables, and operation symbols. They represent relationships and allow us to perform calculations involving unknown quantities.
In algebra, an expression can be a single term or a combination of multiple terms connected by plus or minus signs.
Consider our original expression, \((5a^4 - 2b)^2\). This is a binomial because it has two terms: \(5a^4\) and \(-2b\).
To simplify it, we:
  • Use the binomial theorem: \((x - y)^2 = x^2 - 2xy + y^2\).
  • Identify \((x\) and \((y\): \((x = 5a^4\) and \((y = 2b\).
By following these steps, we turn a complex expression into a sum of simpler algebraic terms.
Polynomials
Polynomials are algebraic expressions with one or more terms, each consisting of a variable raised to a non-negative integer power multiplied by a coefficient. They can be simple or very complex.
The polynomial resulting from our exercise is \((25a^8 - 20a^4b + 4b^2\).
  • \((25a^8\) is a term of degree 8.
  • \((-20a^4b\) is a term of degree 5 (since \((a^4b = a^4b^1\)).
  • \((4b^2\) is a term of degree 2.
Polynomials are essential in algebra because they help describe curves, model real-world problems, and simplify complex calculations. Understanding how to form and manipulate them is a fundamental part of mastering algebra.

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Which of the following is not a difference of two squares? Explain. a) \(16 a^{8} y^{4}-25 c^{12}\) b) \(a^{9}-b^{4}\) c) \(t^{90}-1\) d) \(x^{2}-196\)
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