Factor f(x) into linear factors given that k is a zero of f(x).
$f ( x ) = x ^ { 4 } + 6 x ^ { 3 } + 7 x ^ { 2 } - 12 x - 18$ k= -3 (multiplicity 2)
A) $f ( x ) = ( x - 3 ) ( x + 3 ) ( x - \sqrt { 2 } ) ( x + \sqrt { 2 } )$
B) $f ( x ) = ( x + 3 ) ^ { 2 } ( x - \sqrt { 2 } ) ( x + \sqrt { 2 } )$
C) $f ( x ) = ( x + 3 ) ^ { 2 } ( x - 2 ) ( x + 2 )$
D) $f ( x ) = ( x - 3 ) ^ { 2 } ( x - \sqrt { 2 } ) ( x + \sqrt { 2 } )$

Question

Quizplus · Accepted Answer

Since k = -3 is a zero of f(x) with multiplicity 2, the factor $(x + 3)^2$ must be part of the factorization. The remaining factors must multiply to give a quartic polynomial, and option B correctly includes $(x - \sqrt{2})(x + \sqrt{2})$ to complete the factorization into linear factors.

Factor F(x) into Linear Factors Given That K Is a Zero