The polynomial of degree 4, P(x) has a root multiplicity 2 at x=4 and roots multiplicity 1 at x=0 and x=-4 and it goes through the point (5, 18)
how do you find a formula for p(x)?

Question

Benjamin Achtenberg · Accepted Answer

The polynomial is #P(x)=2/5x(x-4)^2(x+4)# If the polynomial has a root of multiplicity 2 at #x=4#, the #(x-4)^2# is a factor Multiplicity 1 at #x=0#, then #x# is a factor Multiplicity 1 at #x=-4#, then #(x+4)# is a factor So #P(x)=Ax(x-4)^2(x+4)# As it pases through #(5,18)# so #18=A*5*(5-4)^2*(5+4)# So #A=18/5*1/9=2/5# The polynomial is #P(x)=2/5x(x-4)^2(x+4)#

Liam Brennan · Answer

undefined To find a formula for $ P(x) $, we utilize the fact that when a polynomial has a root of multiplicity $ m $ at $ x = a $, its factorization includes $ (x - a)^m $. Given the information provided, the factorization of $ P(x) $ would be $ (x - 4)^2 \cdot x \cdot (x + 4) $. To determine the coefficient of $ P(x) $, we use the given point $ (5, 18) $ by substituting $ x = 5 $ into the polynomial equation and solving for the coefficient. With this information, we can write the formula for $ P(x) $.

The polynomial of degree 4, P(x) has a root multiplicity 2 at x=4 and roots multiplicity 1 at x=0 and x=-4 and it goes through the point (5, 18) how do you find a formula for p(x)?