Domanda del libro di testo
List all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers. {-11, -5/6, 0, 0.75, √5, π, √64}
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Ch. P - Fundamental Concepts of Algebra
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 1, Problema 38
Factor each trinomial, or state that the trinomial is prime.
Guida verificata passo dopo passo1
Identify the trinomial to factor: \(6x^{2} - 7xy - 5y^{2}\).
Multiply the coefficient of \(x^{2}\) (which is 6) by the coefficient of \(y^{2}\) (which is -5), giving \(6 \times (-5) = -30\).
Find two numbers that multiply to -30 and add to the middle coefficient, which is -7. These numbers are -10 and 3 because \(-10 \times 3 = -30\) and \(-10 + 3 = -7\).
Rewrite the middle term \(-7xy\) as \(-10xy + 3xy\) to split the trinomial: \(6x^{2} - 10xy + 3xy - 5y^{2}\).
Group the terms in pairs and factor each group: from \(6x^{2} - 10xy\), factor out \$2x$ to get \(2x(3x - 5y)\); from \(3xy - 5y^{2}\), factor out $y$ to get $y(3x - 5y)$. Then factor out the common binomial $(3x - 5y)$ to get $(3x - 5y)(2x + y)$.
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Factoring Trinomials
Factoring trinomials involves expressing a quadratic expression as a product of two binomials. For trinomials in two variables, such as 6x²−7xy−5y², the goal is to find two binomials whose product equals the original expression. This process often requires identifying pairs of terms that multiply to the constant term and add to the middle term.
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Factor Using Special Product Formulas
Prime Trinomials
A trinomial is prime if it cannot be factored into the product of two binomials with integer coefficients. Determining whether a trinomial is prime involves attempting to factor it and, if no suitable factors exist, concluding that it is prime. Recognizing prime trinomials helps avoid unnecessary factoring attempts.
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05:20Expanding Radicals
Factoring by Grouping
Factoring by grouping is a method used when the trinomial's leading coefficient is not 1. It involves splitting the middle term into two terms whose coefficients multiply to the product of the leading coefficient and the constant term. Then, grouping terms and factoring out common factors from each group can lead to the factorization of the entire trinomial.
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04:36Factor by Grouping
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