If q(x)=x²+5x+6 is a factor of p(x)=kx³+4x²+rx-5, find :(a) the value of the constants k and r
(b) the reminder when p(x) is divided by x-2.
Answers
If q(x) =x² + 5x + 6 is a factor of p(x) = kx3 + 4x2 + rx - 6, what are the values of the constant k and r and the remainder when p(X) is divided by x - 2?
Note that q(x) = (x + 2)(x + 3). If q(x) is a factor of p(x) then the factors of q(x) are also factors of p(x), and then you can apply the Factor Theorem; the values of x that make those factors zero will also make p(x) zero, so p(-2) is zero and so is p(-3).
Substituting x = -2 and x = -3 separately into p(x) will give you two equations, -8k + 16 - 2r - 6 = 0 and another similar equation. Tidy up. You now have two equations and two unknowns, k and r, and since the equations are non-degenerate you can find out k and r and so rewrite p(x) as a cubic function of x.
Finally apply the Remainder Theorem to the resulting equation: the remainder when p(x) is divided by x - 2 is the value of p(x) when x = 2.
Step-by-step explanation:
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