Problem 92 - Algebra and Trigonometry

Question

Factor completely, or state that the polynomial is prime. $$2 x^{3}-98 a^{2} x+28 x^{2}+98 x$$

Step-by-Step Solution

Answer

The complete factorization of the polynomial is \(2x(x+7a)(x-7a)(2x+7)\).

Step 1: Identify Common Factors

Look at each term in the polynomial and identify any common factors. The given polynomial is \(2 x^{3}-98 a^{2} x+28 x^{2}+98 x\). Considering all terms, the common factor among them is \(2x\).

Step 2: Factor out the Common Factor

Next, factor out the common factor from each term in the polynomial. This will leave us with \(2x(x^{2} - 49a^{2} + 14x + 49)\).

Step 3: Factor the Remaining Polynomial

The remaining polynomial inside the parenthesis can be factored further. Separate the terms in sets of two to simplify further. This gives us \(2x[(x^{2} - 49a^{2}) + (14x + 49)]\). Looking at first part, we can see that it's a difference of squares which can be factored into \( (x+7a)(x-7a) \), and the second part can be factored out by 7, which gives \(7(2x+7)\), thus the complete factorization of the polynomial is \(2x(x+7a)(x-7a)(2x+7)\).