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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 89
Factor completely, or state that the polynomial is prime.
Guida verificata passo dopo passo1
First, identify the greatest common factor (GCF) of the terms in the polynomial \(15x^3 + 3x^2\). Look at the coefficients (15 and 3) and the variable parts (\(x^3\) and \(x^2\)) separately.
The GCF of the coefficients 15 and 3 is 3. For the variable parts, the smallest power of \(x\) common to both terms is \(x^2\). So, the overall GCF is \$3x^2$.
Factor out the GCF \$3x^2$ from each term in the polynomial. This means rewriting the polynomial as \(3x^2(\text{something})\) where the 'something' is what remains after dividing each term by \$3x^2$.
Divide each term by \$3x^2$: \(\frac{15x^3}{3x^2} = 5x\) and \(\frac{3x^2}{3x^2} = 1\). So, the expression inside the parentheses is \(5x + 1\).
Write the completely factored form as \(3x^2(5x + 1)\). Since \(5x + 1\) cannot be factored further, this is the complete factorization.
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Greatest Common Factor (GCF)
The Greatest Common Factor is the largest expression that divides all terms of a polynomial without leaving a remainder. Factoring out the GCF simplifies the polynomial and is often the first step in factoring. For example, in 15x^3 + 3x^2, the GCF is 3x^2.
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Graphs of Common Functions
Factoring Polynomials
Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or factors. This process helps in solving equations and simplifying expressions. After factoring out the GCF, check if the remaining polynomial can be factored further.
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07:30Introduction to Factoring Polynomials
Prime Polynomials
A prime polynomial is one that cannot be factored further over the set of integers. After attempting to factor out the GCF and other methods, if no factors are found, the polynomial is considered prime. Recognizing prime polynomials prevents unnecessary factoring attempts.
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07:30Introduction to Factoring Polynomials
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