Textbook Question
In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication.x² + 8x + 12
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0. Review of Algebra
Factoring Polynomials
2:54 minutesProblem 6
Textbook Question
Work each problem. Match each polynomial in Column I with its factored form in Column II.
Verified step by step guidance1
Identify the type of each polynomial in Column I by recognizing patterns such as perfect square trinomials or difference of squares.
For polynomial a: \(x^2 + 10xy + 25y^2\), check if it fits the form of a perfect square trinomial \(a^2 + 2ab + b^2 = (a+b)^2\). Here, \(a = x\) and \(b = 5y\).
For polynomial b: \(x^2 - 10xy + 25y^2\), check if it fits the form of a perfect square trinomial \(a^2 - 2ab + b^2 = (a-b)^2\). Again, \(a = x\) and \(b = 5y\).
For polynomial c: \(x^2 - 25y^2\), recognize it as a difference of squares, which factors as \(a^2 - b^2 = (a+b)(a-b)\) with \(a = x\) and \(b = 5y\).
For polynomial d: \$25y^2 - x^2\(, also a difference of squares but with reversed terms, factor it similarly as \)a^2 - b^2 = (a+b)(a-b)\( where \)a = 5y\( and \)b = x$.
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2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factoring Perfect Square Trinomials
A perfect square trinomial is a quadratic expression that can be written as the square of a binomial, such as (x + a)^2 = x^2 + 2ax + a^2. Recognizing these allows you to factor expressions like x^2 + 10xy + 25y^2 into (x + 5y)^2 efficiently.
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Difference of Squares
The difference of squares formula states that a^2 - b^2 = (a + b)(a - b). This concept helps factor expressions like x^2 - 25y^2 into (x + 5y)(x - 5y), by identifying the two terms as perfect squares separated by subtraction.
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Matching Polynomials to Factored Forms
Matching involves recognizing the structure of polynomials and associating them with their equivalent factored expressions. This requires understanding factoring patterns and comparing terms to select the correct binomial factors from given options.
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