Question - Simplifying Algebraic Expression by Factoring
Solution:
Certainly! To simplify the given expression, we need to factor out common terms and potentially simplify both the numerator and the denominator. The expression is:(10x^(5n+4) + 125x^(5n+2)) / (3x^(5n+3) - 20x^(5n+1))Now, let's simplify both parts:The numerator can be factored by x^(5n+2), which is the smallest power of x present in both terms:x^(5n+2)(10x^2 + 125)Similarly, the denominator can be factored by x^(5n+1):x^(5n+1)(3x^2 - 20)After factoring out, we get:x^(5n+2)(10x^2 + 125) / x^(5n+1)(3x^2 - 20)Now, we can reduce the expression by eliminating common factors. Notice that x^(5n+2) in the numerator and x^(5n+1) in the denominator can be simplified:x^(5n+2) / x^(5n+1) = x ((5n+2) - (5n+1)) = x^(1)Now we have:x(10x^2 + 125) / (3x^2 - 20)This cannot be further simplified without knowing the specific values of x or unless the quadratic expressions factor into terms that cancel each other out (which is not the case here). Thus the simplified expression is:x(10x^2 + 125) / (3x^2 - 20)
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