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Chapter 1: Problem 82

Factor the expression completely. $$49-4 y^{2}$$

Short Answer

Expert verified
The expression \(49 - 4y^2\) factors to \((7 - 2y)(7 + 2y)\).

Step by step solution

01

Recognize the Form

The expression \(49 - 4y^2\) is in the form of \(a^2 - b^2\), which is known as a difference of squares. This type of expression can be factored using the identity: \(a^2 - b^2 = (a - b)(a + b)\).
02

Identify \(a^2\) and \(b^2\)

In the expression \(49 - 4y^2\), identify \(a^2 = 49\) and \(b^2 = 4y^2\). Recognize that \(49\) is a perfect square, so \(a = 7\). For \(4y^2\), we also have a perfect square, so \(b = 2y\).
03

Apply the Difference of Squares Formula

Using the values of \(a\) and \(b\) found in Step 2, apply the difference of squares formula: \((a - b)(a + b)\). Substitute \(a = 7\) and \(b = 2y\) into the formula: \((7 - 2y)(7 + 2y)\).
04

Verify the Factorization

Expand the factored expression \((7 - 2y)(7 + 2y)\) to verify correctness: \(7 \times 7 + 7 \times 2y - 2y \times 7 - 2y \times 2y\). This simplifies to \(49 - 4y^2\), confirming the factorization is correct.

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Key Concepts

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

Difference of Squares
The difference of squares is a special formula used in algebra to simplify expressions. It relies on the identity that states:
  • Given any two terms, where both terms are perfect squares, their difference can be factorized using the formula: \[a^2 - b^2 = (a - b)(a + b)\]
  • This formula is incredibly useful because it reduces a binomial into a product of two binomials.
In the original exercise, the expression \(49-4y^2\) is given. By recognizing that both \(49\) and \(4y^2\) are perfect squares, you can easily apply the difference of squares formula. For instance, \(49\) is the square of \(7\) and \(4y^2\) is the square of \(2y\). Substituting these into the formula, we get \((7 - 2y)(7 + 2y)\). This is a brilliant example of how recognizing the form of an expression can lead to a straightforward factorization using algebraic identities.
Perfect Squares
A perfect square is an expression that is obtained by multiplying a number or an algebraic term by itself. In the world of polynomials and algebra, understanding perfect squares is crucial.
  • For example, \(a^2\) is a perfect square because it is the result of multiplying \(a\) by itself.
  • Similarly, \(b^2\) is a perfect square, derived from \(b \times b\).
In the given exercise, recognizing \(49\) as \(7^2\) and \(4y^2\) as \((2y)^2\) is integral to identifying them as perfect squares. This identification allows you to use different factorization techniques, like the difference of squares formula. Perfect squares often simplify the process by providing clear numeric terms that easily fit back into factorization formulas, helping solve complex problems in more straightforward ways.
Polynomial Factoring
Polynomial factoring is the process of expressing a polynomial as the product of its factors. It's a fundamental tool in algebra that simplifies expressions and solves equations.
  • One of the most common methods is to recognize patterns such as the difference of squares or perfect squares within an expression.
  • These patterns allow for immediate simplification through well-established identities.
The original exercise demonstrates polynomial factoring through the expression \(49 - 4y^2\). By identifying the difference of squares pattern, the expression can be neatly factored into \((7 - 2y)(7 + 2y)\). Factoring is not only about breaking down expressions but also understanding the underlying structure of these expressions. Whether it's solving quadratic equations or simplifying polynomial expressions, mastering factors opens doors to a deeper comprehension of mathematical problems.

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Most popular questions from this chapter

Use a Special Factoring Formula to factor the expression. $$(x+3)^{2}-4$$

Factor the expression completely. Begin by factoring out the lowest power of each common factor. $$\left(x^{2}+1\right)^{1 / 2}+2\left(x^{2}+1\right)^{-1 / 2}$$

Factor the expression completely. $$x^{3}+2 x^{2}+x$$

Factor the expression completely. (This type of expression arises in calculus when using the "product rule.") $$\left(x^{2}+3\right)^{-1 / 3}-\frac{3}{3} x^{2}\left(x^{2}+3\right)^{-4 / 3}$$

Complete the squares in the general equation \(x^{2}+a x+y^{2}+b y+c=0\) and simplify the result as much as possible. Under what conditions on the coefficients \(a, b,\) and \(c\) does this equation represent a circle? A single point? The empty set? In the case that the equation does represent a circle, find its center and radius.
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