p(x) = (3x² - 5)(x + k) - 20 In the polynomial p(x) defined above, k is a constant. If x is a factor of p(x), what is the value of k? A. -6 B. -4 C. 2 D. The graph opens downward and has two positive x-intercepts.
Answer (1)
To find the value of k such that x is a factor of the polynomial p ( x ) = ( 3 x 2 − 5 ) ( x + k ) − 20 , we can use the fact that for x to be a factor of a polynomial, the polynomial must be equal to zero when x = 0 .
Let's substitute x = 0 into the equation for p ( x ) :
Substitute x = 0 into 3 x 2 − 5 to obtain 3 ( 0 ) 2 − 5 = − 5 .
For the term ( 3 x 2 − 5 ) ( x + k ) with x = 0 , this becomes: ( − 5 ) ( k ) = − 5 k .
Substitute x = 0 into the complete expression: p ( 0 ) = − 5 k − 20 .
Set p ( 0 ) = 0 since x = 0 is a factor: − 5 k − 20 = 0 .
Solve for k : − 5 k = 20 .
Divide both sides by − 5 : k = − 4 .
Therefore, the value of k is − 4 .
The correct answer is B. -4 .
