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Chapter 1: Problem 8

Express as a polynomial. $$(5 x-4 y)^{2}$$

Short Answer

Expert verified
The polynomial is \( 25x^2 - 40xy + 16y^2 \).

Step by step solution

01

Identify the Binomial

The expression given is \((5x - 4y)^2\), which is a binomial squared. A binomial is an algebraic expression containing two distinct terms.
02

Apply the Binomial Square Formula

The square of a binomial \((a - b)^2\) is given by the formula \((a^2 - 2ab + b^2)\). Here, identify \(a = 5x\) and \(b = 4y\).
03

Calculate Each Term

Using the formula, calculate each part: 1. \(a^2 = (5x)^2 = 25x^2\).2. \(b^2 = (4y)^2 = 16y^2\).3. The middle term \(-2ab = -2(5x)(4y) = -40xy\).
04

Combine the Terms into a Polynomial

Combine all terms to express the expanded polynomial: \[ 25x^2 - 40xy + 16y^2 \].

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Key Concepts

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

Binomial Expansion
Binomial expansion is a technique used to expand expressions that are raised to a power. It's especially useful when dealing with algebraic expressions in the form
  • (a + b)^n
where "a" and "b" are terms in the binomial. The expansion involves breaking down the expression into a sum of terms, each of which involves a power of "a" and "b".
The binomial theorem provides a systematic way to expand the expression. But for simple cases, like squaring a binomial, special formulas can be applied directly to simplify the process. In our example,
  • (5x - 4y)^2
we specifically use the binomial square formula, which provides a straightforward path to expansion without resorting to the full binomial theorem.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations that represent a particular value or relationship. They are fundamental to understanding equations and mathematical reasoning.
Essential elements of algebraic expressions include:
  • Terms: These are the distinct parts of an expression involving numbers and variables, like 5x or 4y.
  • Coefficients: The numerical parts of terms, such as "5" in 5x, indicating how many times the variable is multiplied.
  • Operators: Symbols that show the operations to be performed between terms, like "+" or "-".
In a binomial, the structure is straightforward with only two distinct terms, as seen in
  • (5x - 4y)
Understanding the parts of an algebraic expression is key to recognizing how they behave when manipulated, such as during expansion processes.
Binomial Square Formula
The binomial square formula is a handy tool for quickly expanding binomials that are squared. It states:
  • (a + b)^2 = a^2 + 2ab + b^2
  • (a - b)^2 = a^2 - 2ab + b^2
This formula provides a direct computation method, helping you avoid tedious multiplication steps. In the example
  • (5x - 4y)^2
we identify "a" as 5x and "b" as 4y. By substituting into the formula, you compute:
  • (5x)^2 = 25x^2
  • (4y)^2 = 16y^2
  • -2(5x)(4y) = -40xy
Therefore, the polynomial expansion is summed up as
  • 25x^2 - 40xy + 16y^2
The binomial square formula streamlines the expansion process, ensuring accuracy and reducing computational efforts in polynomial expressions.

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Most popular questions from this chapter

Simplify the expression. $$\left(2 x^{2}-3 x+1\right)(4)(3 x+2)^{3}(3)+(3 x+2)^{4}(4 x-3)$$

Simplify the expression, and rationalize the denominator when appropriate. $$\sqrt[6]{\left(7 u^{-3} v^{4}\right)^{6}}$$

In evaluating negative numbers raised to fractional powers, it may be necessary to evaluate the root and integer power separately. For example, \((-3)^{2 / 5}\) can be evaluated successfully as \(\left[(-3)^{1 / 5}\right]^{2}\) or \(\left[(-3)^{2}\right]^{1 / 5}\), whereas an error message might otherwise appear. Approximate the realnumber expression to four decimal places. (a) \(\sqrt{\pi+1}\) (b) \(\sqrt[3]{17.1}+5^{14}\)

Simplify the expression. $$\frac{3 t}{t+2}+\frac{5 t}{t-2}-\frac{40}{t^{2}-4}$$

Simplify the expression. $$\frac{5 x}{2 x+3}-\frac{6}{2 x^{2}+3 x}+\frac{2}{x}$$
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