Mathematics

Mathematics

Problem

A polynomial of least possible degree has x-intercepts at x=−3x=-3 and x=3x=3, is symmetric about the y-axis, and passes through the point (0,18)(0,18). Which of the following is its equation? a) f(x)=x2−18f(x)=x^{2}-18 b) f(x)=2 (x2−9)f(x)=2\,(x^{2}-9) c) f(x)=−2 (x2−9)f(x)=-2\,(x^{2}-9) d) f(x)=−12 (x2−9)f(x)=-\dfrac{1}{2}\,(x^{2}-9)
Intuitive explanation

The key insight is recognizing that y-axis symmetry means the polynomial is even, containing only even powers of xx. Since you need x-intercepts at x=±3x = \pm 3, the simplest even polynomial with these roots is (x2−9)(x^2 - 9). The challenge is finding the right constant multiple so the curve passes through (0,18)(0, 18)—just evaluate at that point to lock in the scaling factor.

Intuitive explanation

The key insight is recognizing that y-axis symmetry means the polynomial is even, containing only even powers of xx. Since you need x-intercepts at x=±3x = \pm 3, the simplest even polynomial with these roots is (x2−9)(x^2 - 9). The challenge is finding the right constant multiple so the curve passes through (0,18)(0, 18)—just evaluate at that point to lock in the scaling factor.

Browse the problem library