Mathematics

Mathematics

Problem

Consider the polynomial p(x)=2x3−3x2+4x−5p(x)=2x^{3}-3x^{2}+4x-5. Answer the following questions. a. (i) Find p(2)p(2). b. (ii) Find p(−1)p(-1). c. (iii) Using your results from parts (i) and (ii), determine whether x=2x=2 or x=−1x=-1 is a zero of p(x)p(x). Give reasons for your answer.
Intuitive explanation

This problem is about understanding polynomial structure and behavior. The key insight is that a cubic polynomial's properties—its roots, end behavior, and critical points—are all encoded in its coefficients and derivatives. Rather than trying to solve p(x)=0p(x) = 0 directly (which is hard), you'll use calculus and algebraic techniques to extract useful information about where the function increases, decreases, and changes shape.

Intuitive explanation

This problem is about understanding polynomial structure and behavior. The key insight is that a cubic polynomial's properties—its roots, end behavior, and critical points—are all encoded in its coefficients and derivatives. Rather than trying to solve p(x)=0p(x) = 0 directly (which is hard), you'll use calculus and algebraic techniques to extract useful information about where the function increases, decreases, and changes shape.

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