Textbook Question
In Exercises 91–100, simplify using properties of exponents. (3y1/4)3/y1/12
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Ch. P - Fundamental Concepts of Algebra
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- 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
Chapter 1, Problem 101
Factor completely, or state that the polynomial is prime. 3x^4-12x^2
Verified step by step guidance1
Step 1: Identify the greatest common factor (GCF) of the terms in the polynomial. The terms are 3x⁴ and -12x². The GCF is 3x².
Step 2: Factor out the GCF, 3x², from each term. This gives 3x²(x² - 4).
Step 3: Observe the remaining factor, x² - 4. Notice that it is a difference of squares, which can be factored further using the formula a² - b² = (a - b)(a + b).
Step 4: Apply the difference of squares formula to x² - 4. This results in (x - 2)(x + 2).
Step 5: Combine all the factors. The completely factored form of the polynomial is 3x²(x - 2)(x + 2).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factoring Polynomials
Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or factors. This process is essential for simplifying expressions and solving equations. Common techniques include factoring out the greatest common factor (GCF), using special products like the difference of squares, and applying methods such as grouping or the quadratic formula for higher-degree polynomials.
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Greatest Common Factor (GCF)
The greatest common factor (GCF) is the largest factor that divides two or more numbers or terms without leaving a remainder. In polynomial expressions, identifying the GCF allows for simplification by factoring it out, which can make the remaining polynomial easier to work with. For example, in the polynomial 3x^4 - 12x^2, the GCF is 3x^2.
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Prime Polynomials
A polynomial is considered prime if it cannot be factored into the product of two non-constant polynomials with real coefficients. Recognizing prime polynomials is crucial in algebra, as it indicates that the polynomial cannot be simplified further. In the context of the given polynomial, if after attempting to factor it, no further factorization is possible, it is classified as prime.
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Related Practice
Textbook Question
In Exercises 97–102, write each algebraic expression without parentheses. 1/3(3x)+[(4y)+(−4y)]
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Textbook Question
In Exercises 87–106, perform the indicated computations. Write the answers in scientific notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places. (2.4×10^−2)/(4.8×10^-6)
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Textbook Question
In Exercises 103–110, insert either <, >, or = in the shaded area to make a true statement. |−20| □ |−50|
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Textbook Question
Simplify by reducing the index of the radical.
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Textbook Question
Factor and simplify each algebraic expression.
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