Factor completely $8 x^5+2 x^4+4 x^2$
Answer (2)
Identify the greatest common factor (GCF) of the polynomial terms.
Determine the GCF of the coefficients: 8, 2, and 4, which is 2.
Determine the GCF of the variable terms: x 5 , x 4 , and x 2 , which is x 2 .
Factor out the GCF 2 x 2 from the polynomial: 8 x 5 + 2 x 4 + 4 x 2 = 2 x 2 ( 4 x 3 + x 2 + 2 ) . The final answer is 2 x 2 ( 4 x 3 + x 2 + 2 ) .
Explanation
Problem Analysis We are given the polynomial 8 x 5 + 2 x 4 + 4 x 2 and asked to factor it completely.
Finding GCF of Coefficients First, we look for the greatest common factor (GCF) of the terms in the polynomial. The coefficients are 8, 2, and 4. The GCF of these numbers is 2.
Finding GCF of Variables Next, we look at the variable terms: x 5 , x 4 , and x 2 . The GCF of these terms is x 2 because it is the lowest power of x present in all terms.
Factoring out the GCF Therefore, the GCF of the entire polynomial is 2 x 2 . We factor this out from each term: 8 x 5 + 2 x 4 + 4 x 2 = 2 x 2 ( 4 x 3 + x 2 + 2 )
Checking for Further Factorization Now we consider the cubic polynomial 4 x 3 + x 2 + 2 . We want to see if this can be factored further. Since there are no obvious rational roots, we can assume that it is irreducible over the rationals.
Final Factorization Thus, the complete factorization of the given polynomial is 2 x 2 ( 4 x 3 + x 2 + 2 ) .
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 representing 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 optimize rendering algorithms, making them more efficient and faster.
The polynomial 8 x 5 + 2 x 4 + 4 x 2 is factored completely as 2 x 2 ( 4 x 3 + x 2 + 2 ) . The greatest common factor is 2 x 2 , which is taken out from each term. The remaining cubic polynomial does not factor further using rational numbers.
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