Textbook Question
In Exercises 1–30, factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication.x² − 8x + 15
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0. Review of Algebra
Factoring Polynomials
3:14 minutesProblem 10
Textbook Question
Work each problem. Which of the following is the correct factorization of x3+8?
A. (x+2)3
B. (x+2)(x2+2x+4)
C. (x+2)(x2-2x+4)
D. (x+2)(x2-4x+4)
Verified step by step guidance1
Recognize that the expression \(x^3 + 8\) is a sum of cubes, since \$8\( can be written as \)2^3\(. So, we have \)x^3 + 2^3$.
Recall the formula for factoring a sum of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
Identify \(a = x\) and \(b = 2\) in the formula, then substitute these values into the factorization formula.
Write the factorization as \((x + 2)(x^2 - (x)(2) + 2^2)\), which simplifies to \((x + 2)(x^2 - 2x + 4)\).
Compare this factorization to the given options to determine which one matches the expression \((x + 2)(x^2 - 2x + 4)\).
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sum of Cubes Formula
The sum of cubes formula states that a³ + b³ = (a + b)(a² - ab + b²). This identity helps factor expressions where two cubes are added together, such as x³ + 8, by identifying a = x and b = 2.
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Polynomial Factorization
Polynomial factorization involves rewriting a polynomial as a product of simpler polynomials. Recognizing special forms like sum or difference of cubes allows efficient factorization, which simplifies solving or analyzing polynomial expressions.
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Identifying Correct Factorization
After applying the sum of cubes formula, it is important to match the resulting factors with the given options. This requires careful substitution and verification of each term to ensure the factorization is accurate and corresponds to the original polynomial.
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