Textbook Question
In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication.x² + xy + y²
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0. Review of Algebra
Factoring Polynomials
1:37 minutesProblem 28
Textbook Question
In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication.x² − xy + y²
Verified step by step guidance1
Identify the trinomial: \(x^2 - xy + y^2\).
Check if the trinomial can be factored by looking for two numbers that multiply to \(y^2\) and add to \(-xy\).
Notice that the trinomial resembles a perfect square trinomial, but it does not fit the form \((a - b)^2 = a^2 - 2ab + b^2\).
Since there are no two numbers that satisfy both conditions, conclude that the trinomial is not factorable over the integers.
State that the trinomial \(x^2 - xy + y^2\) is prime.
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1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factoring Trinomials
Factoring trinomials involves rewriting a quadratic expression in the form ax² + bx + c as a product of two binomials. This process requires identifying two numbers that multiply to ac (the product of a and c) and add to b. Understanding this concept is crucial for simplifying expressions and solving equations.
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Prime Trinomials
A prime trinomial is a quadratic expression that cannot be factored into simpler binomial expressions with rational coefficients. Recognizing when a trinomial is prime is essential, as it indicates that no integer solutions exist for the factors. This concept helps in determining the limits of factorization.
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FOIL Method
The FOIL method is a technique used to multiply two binomials, standing for First, Outside, Inside, Last, which refers to the order in which the terms are multiplied. This method is essential for verifying the correctness of a factorization by ensuring that the product of the factors returns to the original trinomial. Mastery of FOIL is important for confirming factorization results.
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