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Ch. P - Fundamental Concepts of Algebra
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. P - Fundamental Concepts of AlgebraProblem 91
Chapter 1, Problem 91

In Exercises 65–92, factor completely, or state that the polynomial is prime. 2x3−8a2x+24x2+72x

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1
Group the terms of the polynomial into two pairs to make factoring easier: \( (2x^3 - 8a^2x) + (24x^2 + 72x) \).
Factor out the greatest common factor (GCF) from each group: \( 2x(x^2 - 4a^2) + 24x(x + 3) \).
Notice that \( x^2 - 4a^2 \) is a difference of squares. Factor it as \( (x - 2a)(x + 2a) \), so the expression becomes \( 2x(x - 2a)(x + 2a) + 24x(x + 3) \).
Factor out the common term \( x \) from the entire expression: \( x[2(x - 2a)(x + 2a) + 24(x + 3)] \).
Simplify the expression inside the brackets and check if further factoring is possible. Combine like terms and verify if the polynomial is fully factored or if it is prime.

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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 its 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 all terms in a polynomial. Identifying the GCF is often the first step in factoring, as it simplifies the polynomial and makes it easier to work with. For example, in the polynomial 2x^3−8a^2 x+24x^2+72x, the GCF can be factored out to simplify the expression before further factoring.
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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, determining whether it is prime or can be factored completely is essential for solving the exercise.
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Related Practice
Textbook Question

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. (6.1×10−8)(2×10−4)(6.1\(\times\)10^{-8})(2\(\times\)10^{-4})

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Textbook Question

Factor completely, or state that the polynomial is prime. x2−11x+28x^2-11x+28

Textbook Question

Factor completely, or state that the polynomial is prime. 64-x^2

Textbook Question

In Exercises 85–96, simplify each algebraic expression. 5(3y−2)−(7y+2)

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Textbook Question

Evaluate each expression without using a calculator. 16-6/2

Textbook Question

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. (4.3×108)(6.2×104)(4.3×10^8)(6.2×10^4)