Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

0. Review of Algebra

Factoring Polynomials

College Algebra

0. Review of Algebra

Factoring Polynomials

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1:37 minutes

Problem 27

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 guidance

Identify the trinomial: \(x^2 + xy + y^2\).

Check if the trinomial can be factored by looking for two binomials \((ax + by)(cx + dy)\) that multiply to give the original trinomial.

Consider the structure of the trinomial: it resembles a perfect square trinomial, which is typically of the form \((x + y)^2 = x^2 + 2xy + y^2\).

Notice that the middle term \(xy\) is not twice the product of \(x\) and \(y\), which suggests that this trinomial might not be a perfect square.

Conclude that \(x^2 + xy + y^2\) is a prime trinomial because it cannot be factored into simpler polynomials with integer coefficients.

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Key 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.

Prime Trinomials

A prime trinomial is a quadratic expression that cannot be factored into the product of two binomials with rational coefficients. Recognizing when a trinomial is prime is essential, as it indicates that the expression cannot be simplified further. This concept helps in determining the nature of the roots of the quadratic equation.

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 also useful for checking the accuracy of a factorization by ensuring that the product of the binomials returns to the original trinomial. Mastery of FOIL is important for verifying solutions in algebra.