Mathematics

Mathematics

Problem

Consider the cubic polynomial p(x)=mx3+3x2+nx−4p(x)=mx^3 + 3x^2 + nx - 4. Given that both x=1x=1 and x=−2x=-2 are zeros of p(x)p(x), find the ordered pair (m,n)(m,n). a) (0,1)(0,1) b) (−3,4)(-3,4) c) (1,0)(1,0) d) (−1,2)(-1,2)
Intuitive explanation

When a polynomial has known zeros, you can use the fact that plugging those values in must give zero—this gives you equations in your unknowns. Here, x=1x=1 and x=−2x=-2 being zeros means p(1)=0p(1)=0 and p(−2)=0p(-2)=0, which creates a system of two linear equations in the two unknowns mm and nn. Solve this system to find the coefficients.

Intuitive explanation

When a polynomial has known zeros, you can use the fact that plugging those values in must give zero—this gives you equations in your unknowns. Here, x=1x=1 and x=−2x=-2 being zeros means p(1)=0p(1)=0 and p(−2)=0p(-2)=0, which creates a system of two linear equations in the two unknowns mm and nn. Solve this system to find the coefficients.

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