Q. Fully simplify the expression below and write your answer as a single fraction.x5−6x4−7x32x2−14x⋅x2−16x+60x2−9x−10Answer:
Identify Given Expression: Identify the given expression and look for common factors in the numerator and denominator that can be simplified.The expression is: (2x2−14x)/(x5−6x4−7x3)×(x2−9x−10)/(x2−16x+60)
Factor Numerator and Denominator: Factor the first numerator and the first denominator.2x2−14x can be factored as 2x(x−7).x5−6x4−7x3 can be factored as x3(x2−6x−7).
Factor Second Numerator and Denominator: Factor the second numerator and the second denominator. x2−9x−10 can be factored as (x−10)(x+1). x2−16x+60 can be factored as (x−10)(x−6).
Rewrite with Factored Terms: Rewrite the expression with the factored terms.x3(x2−6x−7)2x(x−7)⋅(x−10)(x−6)(x−10)(x+1)
Cancel Common Factors: Cancel out the common factors in the numerator and denominator.The (x−10) terms cancel out, and one x term from the first denominator cancels with the x term in the first numerator.The simplified expression is now: x2(x2−6x−7)2(x−7)×x−6x+1
Factor Remaining Quadratic: Factor the remaining quadratic in the denominator. x2−6x−7 can be factored as (x−7)(x+1).
Rewrite with Newly Factored Term: Rewrite the expression with the newly factored term. x2(x−7)(x+1)2(x−7)⋅x−6x+1
Cancel Common Factors: Cancel out the common factors in the numerator and denominator.The (x−7) and (x+1) terms cancel out.The simplified expression is now: x2(x−6)2
Write Final Simplified Expression: Write the final simplified expression as a single fraction.The final answer is: x3−6x22
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