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Polynomials and Rational Functions · Paper 3 — Free Maths AI HL Practice Questions

1ChallengeLAQPolynomial functions and their graphs8 marksPaper 3~12 min
A polynomial function PP of degree nn is defined by P(x)=anxn+an−1xn−1+⋯+a1x+a0,an≠0,  n∈N.P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0, \quad a_n \neq 0,\; n \in \mathbb{N}. Consider f(x)=x4−2x3−7x2+8x+12f(x) = x^4 - 2x^3 - 7x^2 + 8x + 12.
(a)
Show that f(x)=(x+1)(x−2)(x−3)(x+2)f(x) = (x+1)(x-2)(x-3)(x+2). [4 marks]
(b)
The graph of y=f(x)y = f(x) has exactly three turning points. A student claims: "If a polynomial function has degree nn, then its graph has exactly n−1n - 1 turning points." (i) Using the factored form from (a), explain why f(x)f(x) has exactly three turning points. [2]
(ii) Determine whether the student's claim is valid. Justify your answer by constructing a counterexample and stating the correct general result. [4 marks]
diagram

Solutions

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