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Chapter 0: Problem 92

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

Short Answer

Expert verified
The completely factored form of \(2 x^{3}-98 a^{2} x+28 x^{2}+98 x\) is \(x(2x(x + 14) - 98(a + 1)(a - 1))\).

Step by step solution

01

Identify Common Terms

Observing the polynomial \(2 x^{3}-98 a^{2} x+28 x^{2}+98 x\), it's easy to note that every term in the polynomial has x as a common factor. Let's factor out the common term x.
02

Factor out common terms

Factoring out the common term x, we can rewrite the polynomial as \(x(2x^2 + 28x - 98a^2 + 98)\).
03

Reduce the derived factors further

We can observe that \(2x^2 + 28x - 98a^2 + 98\) can further be simplified. Look for common factors and factor them out if possible.
04

Group the terms and factor

Grouping the similar terms, we rewrite this polynomial as \(x(2x(x + 14) - 98(a^2 - 1))\). Notice that \(a^2 - 1\) is a difference of squares, which can be factored as \((a + 1)(a - 1)\).
05

Apply difference of squares

By applying the difference of squares formula, the equation can be further simplified: \(x(2x(x + 14) - 98(a + 1)(a - 1))\).

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