Chapter 4: Problem 35
Factor the polynomial completely. $$ 16 z^4-81 $$
Short Answer
Expert verified
The completely factored form of the polynomial \(16z^4 - 81\) is \((4z^2 + 9)(2z+3)(2z-3)\)
Step by step solution
01
Rewrite the polynomial in the difference of squares form
Rewrite the given polynomial \(16z^4 - 81\) as \((4z^2)^2 - 9^2\)
02
Factor using the difference of squares formula
The difference of squares formula is \((a^2 - b^2) = (a+b)(a-b)\). We can rewrite our expression as \((4z^2 + 9)(4z^2 - 9)\)
03
Notice that you have the difference of squares again
The polynomial \(4z^2 - 9\) is again a difference of squares. Rewrite it as \((2z)^2 - 3^2\)
04
Factor again using the difference of squares formula
The difference of squares formula is \((a^2 - b^2) = (a+b)(a-b)\). The expression \(4z^2 - 9\) can be rewritten as \((2z+3)(2z-3)\)
05
Write your final expression
After completely factoring the difference of squares we get \((4z^2 + 9)(2z+3)(2z-3)\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Difference of Squares
The concept of the difference of squares is a crucial factorization technique in algebra, especially when dealing with polynomials. This technique applies when you have an expression of the form \( a^2 - b^2 \). The notable property of this difference is that it can be expressed as \( (a+b)(a-b) \). This helps simplify and factor otherwise complex algebraic expressions quickly.
For the exercise \( 16z^4 - 81 \), we recognize it as a difference of squares by identifying two perfect squares, \( (4z^2)^2 \) and \( 9^2 \). This recognition is the first step towards the complete factorization of the polynomial. Understanding this concept makes solving equations simpler and more systematic.
For the exercise \( 16z^4 - 81 \), we recognize it as a difference of squares by identifying two perfect squares, \( (4z^2)^2 \) and \( 9^2 \). This recognition is the first step towards the complete factorization of the polynomial. Understanding this concept makes solving equations simpler and more systematic.
Algebraic Expressions
Algebraic expressions consist of terms combined using addition, subtraction, multiplication, and division. They are the building blocks in algebra and can represent complex relationships. In the case of \( 16z^4 - 81 \), this expression is made up of terms involving the variable \( z \) raised to powers and subtracted from a constant term, forming a polynomial.
When factoring these expressions, understanding their structure is essential. Breaking down terms into factors or recognizing the underlying pattern, such as the difference of squares, simplifies equations, making them easier to solve. Always look for patterns and simplifiable structures within any algebraic expression to assist in solving.
When factoring these expressions, understanding their structure is essential. Breaking down terms into factors or recognizing the underlying pattern, such as the difference of squares, simplifies equations, making them easier to solve. Always look for patterns and simplifiable structures within any algebraic expression to assist in solving.
Quadratic Expressions
Quadratic expressions are polynomials of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. They are fundamental in algebra and commonly appear in factorization problems. While our exercise primarily deals with a quartic expression, it splits into smaller quadratic-like differences that can be factored further.
Once \( 16z^4 - 81 \) is factored to \( (4z^2 + 9)(4z^2 - 9) \), the expression \( 4z^2 - 9 \) itself resembles a quadratic form, specifically a difference of squares. Recognizing these types of expressions within larger polynomials is key to efficient factorization.
Once \( 16z^4 - 81 \) is factored to \( (4z^2 + 9)(4z^2 - 9) \), the expression \( 4z^2 - 9 \) itself resembles a quadratic form, specifically a difference of squares. Recognizing these types of expressions within larger polynomials is key to efficient factorization.
Factorization Techniques
Factorization is a powerful algebraic tool used to break down complex expressions into products of simpler factors. It simplifies solving equations and aids in behavioral analysis of algebraic functions. Methods include, but are not limited to, factoring by grouping, using the quadratic formula, and identifying special product forms like the difference of squares.
For the problem \( 16z^4 - 81 \), we applied the difference of squares technique twice, demonstrating a layered factorization approach. Initially, \( 16z^4 - 81 \) was rewritten using the difference of squares as \( (4z^2)^2 - 9^2 \). Subsequently, further factorization of \( 4z^2 - 9 \) utilized the same technique, breaking it down into \( (2z+3)(2z-3) \). Mastery of such techniques is invaluable for tackling diverse algebraic problems efficiently.
For the problem \( 16z^4 - 81 \), we applied the difference of squares technique twice, demonstrating a layered factorization approach. Initially, \( 16z^4 - 81 \) was rewritten using the difference of squares as \( (4z^2)^2 - 9^2 \). Subsequently, further factorization of \( 4z^2 - 9 \) utilized the same technique, breaking it down into \( (2z+3)(2z-3) \). Mastery of such techniques is invaluable for tackling diverse algebraic problems efficiently.