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Chapter 5: Problem 175

Factor each polynomial completely. $$b^{2} n^{2}-y^{4}$$

Short Answer

Expert verified
The polynomial factors to \[(bn - y^2)(bn + y^2)\].

Step by step solution

01

Identify the form

Recognize that the polynomial is in the form of a difference of squares: \[a^2 - b^2\]. Here, we have \[b^2 n^2 - y^4\].
02

Express each term as a square

Rewrite each term as a square: \[ (bn)^2 \text{ and } (y^2)^2\]. So, the expression is: \[ (bn)^2 - (y^2)^2\].
03

Apply the difference of squares formula

The difference of squares can be factored using the formula \[a^2 - b^2 = (a - b)(a + b)\]. Here, \(a = bn\) and \(b = y^2\). This gives: \[ (bn - y^2)(bn + y^2)\].

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Key Concepts

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

Difference of Squares
The concept of difference of squares is crucial in factoring certain polynomials.
A difference of squares occurs when you have an expression of the form \[a^2 - b^2\] Here, both terms are perfect squares, and they are subtracted from each other.
This special form can always be factored into: \((a - b)(a + b)\) For instance, if you see an expression like \(9x^2 - 25\) You'd recognize that \(9x^2 = (3x)^2\) and \(25 = 5^2\) Thus, you can factor it to: \((3x - 5)(3x + 5)\) In our original exercise, the polynomial is \(b^2 n^2 - y^4\) This matches the \[a^2 - b^2\] form when rewritten as \[(bn)^2 - (y^2)^2\] This allows us to factor it into \[(bn - y^2)(bn + y^2)\].
Polynomial Expressions
Polynomials are expressions that consist of variables and coefficients, combined using addition, subtraction, and multiplication.
A simple polynomial could be \(3x^2 + 2x - 1\) Each term in a polynomial can be thought of as a variable raised to a power multiplied by a coefficient.
For example, in the term \(3x^2\) '3' is the coefficient, 'x' is the variable, and '2' is the exponent.
Polynomial expressions can be categorized by their degree, which is the highest power of the variable in the expression.
For instance, \(5x^3 - 4x + 7\) is a degree-3 polynomial because the highest exponent is '3'.
In our original exercise, \(b^2 n^2 - y^4\) is a degree-4 polynomial.
Factoring Techniques
Factoring is the process of breaking down a polynomial into simpler 鈥榝actors鈥� that, when multiplied together, give the original polynomial.
There are multiple factoring techniques including:
  • Common Factor: Finding a common factor in all terms and factoring it out.
  • Grouping: Grouping terms in ways that make factoring simpler.
  • Difference of Squares: As we saw, recognising \(a^2 - b^2\) and applying the \(a^2 - b^2 = (a - b)(a + b)\) formula.

In our example, the difference of squares technique is used to factor \(b^2 n^2 - y^4\) by expressing each term as a square and applying the formula: \[(bn - y^2)(bn + y^2)\]. This highlights how different techniques can simplify complex polynomial expressions into more manageable forms.

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Most popular questions from this chapter

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