What is the value of b?

Question

The polynomial function p is defined by p(x)=4x³+bx²+41x+12, where b is a constant. When graphed on a standard plane, p intersects the x-axis at (-0.25,0), (3,0), and (k,0). What is the value of b?

Answer 1

We can use these points to create a factor polynomial.  That factor polynomial is

 

(x + 1/4)(x - 3)(x - k)

 

 

Then expanding this factorization, we obtain

 

(x + 1/4)(x² - kx - 3x + 3k) =

 

x(x² - kx - 3x + 3k) + (1/4)(x² - kx - 3x + 3k) =

 

x³ - kx² - 3x² + 3kx + (1/4)x² - (1/4)kx - (3/4)x + (3/4)k

 

 

Now we group coefficients by variables.

 

x³ + x²(-k - 3 + (1/4)) + x(3k - (1/4)k - (3/4)) + (3/4)k

 

 

Then, equate coefficients using the expanded factor and p(x).

 

x³ :           4 = 4                                             eq1

x²:           4[-k - 3 + (1/4)] = b                        eq2

x:            4[3k - (1/4)k - (3/4)] = 41               eq3

cons:       4[(3/4)k] = 12                                   eq4

 

 

Now use this system of equations to solve for b.  You will need to solve for k from eq4.  Then substitute that value of k into eq2 to solve for b.