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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1:55 minutes

Problem 21

Textbook Question

In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication.x² − x + 7

Verified step by step guidance

Identify the trinomial: \(x^2 - x + 7\).

Check if the trinomial can be factored by looking for two numbers that multiply to the constant term (7) and add to the linear coefficient (-1).

Since 7 is a prime number, the only integer factors are 1 and 7, which do not add up to -1.

Conclude that the trinomial \(x^2 - x + 7\) cannot be factored using integer coefficients.

State that the trinomial is prime because it cannot be factored further over the integers.

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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. If such numbers do not exist, the trinomial is considered prime, meaning it cannot be factored over the integers.

Prime Trinomials

A prime trinomial is a quadratic expression that cannot be factored into simpler binomial expressions with integer coefficients. This occurs when the discriminant (b² - 4ac) is negative or when no integer pairs satisfy the conditions for factoring. Recognizing prime trinomials is essential for determining the limits of factorization.

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 helps verify the correctness of a factorization by ensuring that the product of the binomials returns to the original trinomial. It is a crucial step in confirming the accuracy of the factorization process.