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Chapter 5: Problem 24

Factor each polynomial. SEE EXAMPLE 1. (OB]ECTIVE 1) $$16 z^{2}-9$$

Short Answer

Expert verified
The polynomial \(16z^2 - 9\) factors to \((4z + 3)(4z - 3)\)

Step by step solution

01

Identify the Values of a and b

From the formula \(a^2 - b^2 = (a+b)(a-b)\), we can identify \(a\) and \(b\) from the given polynomial \(16z^2 - 9\). Note that \(16z^2\) could be written as \((4z)^2\) and \(-9\) could be written as \(-3^2\). Therefore, \(a = 4z\) and \(b = 3\) in this instance.
02

Apply the Difference of Squares Formula

Now we plug \(a\) and \(b\) into the formula \(a^2 - b^2 = (a+b)(a-b)\), which gives us \((4z + 3)(4z - 3)\)
03

Write the Final Factored Form

The final factored form of \(16z^2 - 9\) then is \((4z + 3)(4z - 3)\)

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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 algebraic pattern that allows for easy factorization of certain polynomial expressions. It applies when you have a subtraction between two perfect squares. A perfect square is a number that can be expressed as the square of an integer or a polynomial.
  • The general formula for the difference of squares is \( a^2 - b^2 = (a + b)(a - b) \).
  • For example, if you begin with \( 16z^2 - 9 \), you can recognize this as a difference of squares.
  • Each term, \( 16z^2 \) and \( 9 \), is a perfect square because \( (4z)^2 = 16z^2 \) and \( 3^2 = 9 \).
In this problem, identify \( a \) and \( b \) by taking the square root of each term. Here, \( a = 4z \) and \( b = 3 \). By applying the formula, you can write the expression as \((4z + 3)(4z - 3)\). This shows how the difference of squares simplifies expressions and turns them into factors that are easier to work with.
Polynomial Expressions
A polynomial expression is made up of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. They can vary in degree, which refers to the highest power of the variable present.
  • In the given polynomial \( 16z^2 - 9 \), each term is a product of a number and a power of the variable \( z \).
  • The expression \( 16z^2 \) has a degree of 2, as the highest exponent for \( z \) is 2.
Polynomial expressions are foundational in algebra because they allow us to represent complex relationships, perform arithmetic operations, and solve equations. When you encounter polynomial expressions represented as differences of squares, as in this exercise, you can quickly apply the pattern for factoring, making the process efficient.
Algebraic Formulas
Algebraic formulas provide set methods for solving equations and expressions, allowing for the streamlined manipulation and simplification of algebraic problems.
  • The difference of squares formula \( a^2 - b^2 = (a + b)(a - b) \) is a prime example of an algebraic formula used in factorization.
  • This particular formula is a tool that translates a simple pattern into a factorable form without needing complex calculations.
  • By identifying scenarios where algebraic formulas apply, such as spotting perfect square terms, we can simplify the daunting task of factorization to just a few steps.
Using these formulas helps break down polynomials into more manageable expressions, reducing errors and enhancing understanding. It's a matter of recognizing patterns and practicing application, transforming algebraic expressions with ease, as demonstrated in the problem \( 16z^2 - 9 \).

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Most popular questions from this chapter

Factor. If the polynomial is prime, so indicate. $$5 x^{2}+2+x$$

Factor. Write each trinomial in descending powers of one variable, if necessary. If a polynomial is prime, so indicate. $$x^{2}-13-12 x$$

Factor. Write each trinomial in descending powers of one variable, if necessary. If a polynomial is prime, so indicate. $$y^{2}+7+8 y$$

Factor. If the polynomial is prime, so indicate. $$9 p^{2}+1+6 p-q^{2}$$

Factor each expression completely. $$3 a b+9 a-2 b-6$$
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