Factor completely $x^3+4 x^2+8 x+32$
A. $(x+4)(x^2+8)$
B. $(x-4)(x^2-8)$
C. $(x-4)(x^2+8)$
D. $(x+4)(x^2-8)$
Answer (2)
Group the terms: ( x 3 + 4 x 2 ) + ( 8 x + 32 ) .
Factor out common factors: x 2 ( x + 4 ) + 8 ( x + 4 ) .
Factor out the common binomial: ( x + 4 ) ( x 2 + 8 ) .
The complete factorization is ( x + 4 ) ( x 2 + 8 ) .
Explanation
Problem Analysis We are given the cubic polynomial x 3 + 4 x 2 + 8 x + 32 and asked to factor it completely.
Grouping Terms We will use factoring by grouping to factor the polynomial. First, we group the first two terms and the last two terms: ( x 3 + 4 x 2 ) + ( 8 x + 32 ) .
Factoring out Common Factors Next, we factor out the greatest common factor from each group. From the first group, we can factor out x 2 , and from the second group, we can factor out 8: x 2 ( x + 4 ) + 8 ( x + 4 ) .
Factoring out the Binomial Now, we factor out the common binomial factor ( x + 4 ) from both terms: ( x + 4 ) ( x 2 + 8 ) .
Checking for Further Factorization We need to check if x 2 + 8 can be factored further. Since 8 is not a perfect square, and we are looking for real factors, x 2 + 8 cannot be factored further over real numbers. Therefore, the complete factorization is ( x + 4 ) ( x 2 + 8 ) .
Final Answer The complete factorization of the given polynomial is ( x + 4 ) ( x 2 + 8 ) .
Examples
Factoring polynomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures or circuits. Imagine you are designing a bridge, and you have an equation that describes the load distribution. Factoring this equation can help you identify critical points and ensure the bridge's stability. Similarly, in computer graphics, factoring polynomials can help optimize rendering algorithms, making the graphics smoother and more efficient. Factoring also helps in solving many mathematical problems, such as finding the roots of a polynomial, which can represent the equilibrium points in a system.
The polynomial x 3 + 4 x 2 + 8 x + 32 can be factored completely as ( x + 4 ) ( x 2 + 8 ) . Thus, the correct answer choice is A.
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