E-AP01 · An exact remainder controls the estimate
The first three terms of \((1+x)^4\) are retained as \(P(x)=1+4x+6x^2\). Let \(R(x)=(1+x)^4-P(x)\).
Find an exact factored expression for R.
Use P to estimate \(1.025^4\), and find the exact signed remainder and the direction of the error.
Show without calculus that the absolute error is less than \(0.00004\) whenever \(|x|\le0.02\).
Complete-answer checking, worked solutions and indicative written criteria.
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