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Chapter 6: Problem 94

Factor using rational numbers. $$ 5 x+4 y+25 x^{2}-16 y^{2} $$

Short Answer

Expert verified
The expression factors to \((5x + 4y)(5x - 4y + 1)\).

Step by step solution

01

Identify Quadratic Form

The expression given is \(5x + 4y + 25x^2 - 16y^2\). Notice that we have terms like \(25x^2\) and \(-16y^2\) that suggest a quadratic form after rearranging.
02

Rearrange and Recognize Patterns

Rearrange the terms to group them as: \(25x^2 + 5x - 16y^2 + 4y\). This arrangement helps in recognizing it as a quadratic-like expression, where \(25x^2 - 16y^2\) has a semblance to the difference of squares.
03

Factor Each Group

First, factor the quadratic-like terms separately. For \(25x^2 - 16y^2\), recognize it as a difference of squares: \((5x)^2 - (4y)^2\). Factor this as:\[(5x + 4y)(5x - 4y)\]The other terms \(5x + 4y\) already match one part of the factor, simplifying potential combination.
04

Combine Terms

Notice that \(5x + 4y\) appears in both the initially ordered expression and after factoring the difference of squares. We can factor out \(5x + 4y\) from the expression:\[(5x + 4y)(5x - 4y + 1)\]This gives the final factored form.

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Key Concepts

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

Quadratic Expression
A quadratic expression is a specific type of polynomial that includes terms with the variable raised to the power of two, as seen in terms like \(25x^2\). Quadratics are generally in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. The importance of recognizing a quadratic expression lies in its predictability and structure, which allows us to apply specific factoring techniques.
  • Quadratic expressions often appear as part of larger polynomials.
  • Being able to recognize a quadratic form can simplify the process of solving or factoring polynomials.
In the given exercise, after rearranging the terms, we can see a rough resemblance to a quadratic expression. Identifying these structures is the first step in effective polynomial analysis.
Difference of Squares
The difference of squares is a specific pattern that appears frequently in algebra. It's a tool used for factoring expressions that can be written in the form \(a^2 - b^2\). Recognizing this pattern allows us to rewrite the expression as \((a + b)(a - b)\).
  • The pattern is characterized by two squared terms subtracting each other.
  • Commonly identified by terms such as \((5x)^2 - (4y)^2\).
In our example, \(25x^2 - 16y^2\) is such an expression, where \((5x)^2 - (4y)^2\) leads us to factor it into \((5x + 4y)(5x - 4y)\). This form is essential for simplifying expressions and solving equations that would otherwise be complex to handle.
Rearranging Terms
Rearranging terms in an algebraic expression is a valuable strategy, particularly when dealing with complex polynomials. It involves strategically ordering terms to highlight specific patterns or structures. In our example, arranging \(25x^2 + 5x - 16y^2 + 4y\) set the stage for recognizing that part of the expression, \(25x^2 - 16y^2\), is a difference of squares.
  • This process does not change the value of the expression but makes it easier to identify useful patterns.
  • It often serves as a preparatory step before applying more advanced techniques like factoring.
By rearranging terms, we not only simplify the process of finding factors but also increase the clarity and manageability of handling the expression, ultimately leading to a more straightforward solution.

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Most popular questions from this chapter

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 125 p^{3}-64 y^{3} $$

The area of a teacher's desktop is represented by the trinomial \(\left(4 x^{2}+20 x-11\right)\) in. \(2^{2} .\) Factor it to find the polynomials that represent its length and width.

Complete each factorization. $$ \begin{aligned} 6 m^{3}-28 m^{2}+16 m &=2 m\left(3 m^{2}-\square+8\right) \\ &=2 m(3 m-2)(\square-4) \end{aligned} $$

Factor using rational numbers. $$ x(x-y)-y(y-x) $$

Choose the correct method from Sections 6.1, 6.2, or 6.3 to factor each of the following. $$ x^{3}-2 x^{2}+5 x-10 $$
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