Problem 23
Question
Challenge Problems.$$25 a^{4} b^{4}-9$$
Step-by-Step Solution
Verified Answer
The factored form of \(25a^{4}b^{4} - 9\) is \((5a^2b^2 + 3)(5a^2b^2 - 3)\).
1Step 1: Recognize the Difference of Squares
The expression can be identified as a difference of squares. A difference of squares can be factored using the formula: \(a^2 - b^2 = (a+b)(a-b)\). The given expression, \(25a^{4}b^{4} - 9\), can be rewritten as \((5a^2b^2)^2 - (3)^2\).
2Step 2: Apply the Difference of Squares Formula
Using the difference of squares formula to factor \((5a^2b^2)^2 - (3)^2\), derive the factored form: \[(5a^2b^2 + 3)(5a^2b^2 - 3)\].
3Step 3: Examine Each Factor for Further Factoring
Each of the factors from Step 2 is examined to see if they can be factored further. Neither \(5a^2b^2 + 3\) nor \(5a^2b^2 - 3\) is a difference or sum of squares or any other easily factorable expression in integers, so the factoring is complete.
Key Concepts
Polynomial FactoringDifference of Squares FormulaAlgebraic ExpressionsFactorization Techniques
Polynomial Factoring
In algebra, polynomial factoring is equivalent to breaking down a complex expression into simpler multipliers that, when multiplied together, give back the original polynomial. It plays a vital role in simplifying algebraic expressions and solving polynomial equations.
The general rule here is to look for patterns or common factors in the terms of the polynomial. A simple example would be factoring out a common variable, like in the expression: \(x^2 + 2x = x(x + 2)\). This process not only simplifies the expression but also is crucial in finding the zeros of the polynomial function, an essential concept in calculus and other areas of mathematics.
The general rule here is to look for patterns or common factors in the terms of the polynomial. A simple example would be factoring out a common variable, like in the expression: \(x^2 + 2x = x(x + 2)\). This process not only simplifies the expression but also is crucial in finding the zeros of the polynomial function, an essential concept in calculus and other areas of mathematics.
Difference of Squares Formula
One of the most fundamental patterns in polynomial factoring is identified with the difference of squares formula. It states that any expression that can be written as a square of one term minus the square of another term can be factored into the product of the sum and difference of the two terms. Mathematically, this is expressed as: \(a^2 - b^2 = (a+b)(a-b)\).
For instance, take the expression \(x^2 - 9\). This is a difference of squares as it can be rewritten as \(x^2 - 3^2\), and further factored to \(x + 3)(x - 3)\). Using this technique allows for quick and efficient factorization of applicable quadratic expressions.
For instance, take the expression \(x^2 - 9\). This is a difference of squares as it can be rewritten as \(x^2 - 3^2\), and further factored to \(x + 3)(x - 3)\). Using this technique allows for quick and efficient factorization of applicable quadratic expressions.
Algebraic Expressions
An algebraic expression is a mathematical phrase that includes numbers, variables, and arithmetic operations. For example, \(3x^2 - 2x + 5\) is an algebraic expression with the variable \(x\). Algebraic expressions are the raw materials that make up algebraic equations, and manipulating them correctly is essential for solving those equations.
Understanding the structure of these expressions, the order of operations, and the properties of real numbers helps in their manipulation. Polynomials are specific types of algebraic expressions with variables raised to whole number exponents and combined using addition, subtraction, and multiplication, but not division by variables.
Understanding the structure of these expressions, the order of operations, and the properties of real numbers helps in their manipulation. Polynomials are specific types of algebraic expressions with variables raised to whole number exponents and combined using addition, subtraction, and multiplication, but not division by variables.
Factorization Techniques
Several factorization techniques are taught in algebra to break down more complex expressions. These include pulling out the greatest common factor, using the difference of squares, factoring by grouping, and factoring trinomials, to name a few.
Each technique has specific scenarios where it applies best. For example, the greatest common factor method is used when each term in the polynomial shares a common factor. On the other hand, factoring by grouping is efficient for polynomials with four or more terms, while the difference of squares, as previously discussed, works for binomials that are squares subtracted from each other. Mastery of these techniques greatly enhances a student's ability to simplify algebraic expressions and solve complex equations.
Each technique has specific scenarios where it applies best. For example, the greatest common factor method is used when each term in the polynomial shares a common factor. On the other hand, factoring by grouping is efficient for polynomials with four or more terms, while the difference of squares, as previously discussed, works for binomials that are squares subtracted from each other. Mastery of these techniques greatly enhances a student's ability to simplify algebraic expressions and solve complex equations.
Other exercises in this chapter
Problem 23
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