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Chapter 6: Problem 95

Factor. $$ -169+25 x^{2} $$

Short Answer

Expert verified
The expression \(-169 + 25x^2\) factors to \((5x - 13)(5x + 13)\).

Step by step solution

01

Recognize the Structure

Observe the expression \(-169 + 25x^2\). It is formatted as a difference of squares, where the expression matches the general form \(a^2 - b^2\), where \(a = 5x\) and \(b = 13\).
02

Apply the Difference of Squares Formula

Use the difference of squares formula, \(a^2 - b^2 = (a - b)(a + b)\), to factor the expression.In this context, it's \((5x)^2 - 13^2\). Applying the formula, the expression becomes:\((5x - 13)(5x + 13)\).
03

Verify the Factorization

Multiply the factors \((5x - 13)(5x + 13)\) to ensure they equal the original expression.Use the distributive property:1. \(5x imes 5x = 25x^2\)2. \(5x imes 13 = 65x\)3. \(-13 imes 5x = -65x\)4. \(-13 imes 13 = -169\)Combine the terms: \(25x^2 + 65x - 65x - 169 = 25x^2 - 169\).This verifies 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 a method used in algebra to simplify expressions or solve equations. When you factor a polynomial, you are essentially breaking it down into simpler "factor" expressions that multiply together to produce the original polynomial.
  • Think of it like undoing the multiplication you would have done to originally get the polynomial.
  • A common situation in factoring is finding polynomials that match the pattern of a difference of squares, such as our example, where the expression is \[ -169 + 25x^2 \]
In this particular case, recognizing it as a difference of squares allows you to express it in the form \( a^2 - b^2 \), where \( a = 5x \) and \( b = 13 \).
By identifying such patterns, you can apply specific formulas like the difference of squares pattern \[ (a-b)(a+b) \], to factor the expression easily and efficiently.
Algebraic Expressions
An algebraic expression like \( -169 + 25x^2 \) represents mathematical phrases that include numbers, variables, and operations. These expressions are foundational in algebra and are used to describe equations, reflect relationships, and solve problems.
  • In the expression \( -169 + 25x^2 \), \( -169 \) is a constant term, and \( 25x^2 \) is a term involving a variable raised to a power.
  • Expressions can often be simplified or factored to help in solving algebraic equations, which involves finding the value of the variable that makes the expression true.
Recognizing patterns, such as a difference of squares within algebraic expressions, is crucial. It allows quick simplification through factorization, making complex problems manageable.
Distributive Property
The distributive property is a fundamental principle in algebra used to multiply a single term across terms within a parentheses. Mathematically, it is expressed as \( a(b + c) = ab + ac \).
  • In the context of verifying factored expressions, after applying the difference of squares formula, we used the distributive property to verify our result of \((5x - 13)(5x + 13)\).
  • By expanding \((5x - 13)(5x + 13)\) using the distributive property, each term inside the parentheses is multiplied by each other, ensuring that when collected, they equal the original expression \(-169 + 25x^2\).
Practically, the distributive property ensures accuracy in both multiplication and addition processes, making it indispensable for checking your work in polynomial factoring.

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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. $$ 27 x-27 y-27 z $$

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

Fill in the blanks. A polynomial is factored _____ when each factor is prime.

Solve each equation. $$ 2 a^{3}+a^{2}-32 a-16=0 $$

The following is a list of random factoring problems. Factor each expression. If an expression is not factorable, write "prime." See Examples 1-5. $$ 16 d^{2}-56 d z+49 z^{2} $$
See all solutions

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