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Chapter 6: Problem 7

Factor completely. If the polynomial is not factorable, write prime. $$ 16 w^{2}-169 $$

Short Answer

Expert verified
The polynomial factors as \((4w - 13)(4w + 13)\).

Step by step solution

01

Recognize the Format

Notice that the given polynomial is in the form of a difference of squares, which is a special type of polynomial that can be factored using the formula \(a^2 - b^2 = (a - b)(a + b)\). In this exercise, we need to identify \(a^2\) and \(b^2\).
02

Identify the Squares

The expression \(16w^2\) is a perfect square where \(a^2 = (4w)^2\). Similarly, \(169\) is a perfect square where \(b^2 = 13^2\). So, \(a = 4w\) and \(b = 13\).
03

Apply Difference of Squares Formula

Using the difference of squares formula \(a^2 - b^2 = (a - b)(a + b)\), substitute \(a = 4w\) and \(b = 13\). The expression \(16w^2 - 169\) factors into \((4w - 13)(4w + 13)\).
04

Verify the Factorization

To ensure our factorization is correct, multiply the factors \((4w - 13)(4w + 13)\) using the distributive property (FOIL method): \(4w \cdot 4w + 4w \cdot 13 - 13 \cdot 4w - 13 \cdot 13 = 16w^2 - 169\). This confirms that the factorization is correct.

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Key Concepts

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

Factoring Polynomials
Factoring polynomials is an essential skill in algebra that involves breaking down a polynomial into simpler terms, or factors, that when multiplied together yield the original polynomial. This process is crucial for solving polynomial equations and simplifying expressions.
  • Recognize standard formats: Before you can factor a polynomial, you need to identify its structure. Is it a simple trinomial? A difference of squares? Each type has its own factoring strategy.
  • Factor completely: Always aim to factor a polynomial entirely. Sometimes this involves recognizing smaller degrees of polynomials within the expression. Continuously break it down until no further simple factors can be identified.
  • Use special formulas: There are formulas like the difference of squares formula that accelerate the process. For example, if you detect a polynomial is configured as a difference of squares, you can directly apply \(a^2 - b^2 = (a - b)(a + b)\).
Factoring isn't only a procedural task but also requires intuition and pattern recognition to recognize the format quickly.
Perfect Squares
Perfect squares are integral when factoring polynomials, especially in expressions that involve the difference of squares. A perfect square is a number or expression that can be expressed as the square of another number or expression.
  • Identification: To use perfect squares in factoring, recognize expressions like \(16w^2\) or \(169\). Notice 16 and 169 are perfect squares because \(16 = 4^2\) and \(169 = 13^2\).
  • Use in differences: When dealing with the difference of squares, like in the given problem, identify the perfect squares first. This simplifies the factorization process.
  • Applications beyond: Understanding perfect squares extends beyond just algebra. In geometry, perfect square concepts apply in areas like calculating areas of square shapes.
By mastering the identification and application of perfect squares, breaking down complex polynomials becomes significantly easier.
Polynomial Factorization
Polynomial factorization involves expressing a polynomial as a product of its factors. This can simplify solving equations and understanding polynomial behavior.
  • Difference of Squares: A specific case in polynomial factorization is the difference of squares, which involves identifying \(a^2 - b^2\) patterns. By knowing \(a\) and \(b\), you can quickly factor the polynomial using \((a - b)(a + b)\).
  • Verification step: Once you factor a polynomial, it鈥檚 crucial to verify by multiplying the factors back together to ensure they yield the original polynomial. This is a good practice to confirm accuracy.
  • Connection to real-world problems: Polynomial factorization isn鈥檛 solely an academic exercise. Engineers and scientists often use these methods in areas as diverse as structural analysis and signal processing.
Through consistent practice and understanding of polynomial identities, factorizing polynomials becomes a valuable tool in both mathematical theory and practical applications.

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Most popular questions from this chapter

For Exercises \(11-18,\) complete each of the following. a. Graph each function by making a table of values. b. Determine the consecutive integer values of \(x\) between which each real zero is located. C. Estimate the \(x\) -coordinates at which the relative maxima and relative minima occur. $$ f(x)=x^{4}-10 x^{2}+9 $$

Solve each matrix equation or system of equations by using inverse matrices. $$ \begin{array}{l}{3 j+2 k=8} \\ {j-7 k=18}\end{array} $$

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. $$ x^{3}-6 x^{2}+11 x-6 ; x-2 $$

Factor completely. If the polynomial is not factorable, write prime. $$ 7 x y^{3}-14 x^{2} y^{5}+28 x^{3} y^{2} $$

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. $$ x^{4}+2 x^{3}+2 x^{2}-2 x-3 ; x+1 $$
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