SOLUTION: Suppose that a cubic function f(x)=ax^3+bx^2+cx+d can be factored into two terms,one of which is (x-r). Find the other term using (synthetic division) and answer the questions bel

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Question 935882: Suppose that a cubic function f(x)=ax^3+bx^2+cx+d can be factored into two terms,one of which is (x-r). Find the other term using (synthetic division) and answer the questions below.
(1) ax^3+bx^2+cx+d = (x-r) ( ..........)
(2) what must be true about ar^3 + br^2 +cr +d ? ...
(3 what does this tell us about the value of f(r)? ...
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
answer to question 1:

ax^3 + bx^2 + cx + d = (x-r) * ( ax^2 + (ra+b)x + (r^2a+rb+c) )

answer to question 2:

ar^3 + br^2 + cr + d must be equal to 0

this is because r is a root of the equation which means that the value of f(r) must be equal to 0.

answer to question 3:

value of f(r) must be equal to 0.

this is because f(r) is equal to ar^3 + br^2 + cr + d

when r is a root, f(r) will be equal to 0.

r is a root when the remainder of the division by (x-r) is equal to 0.

when r is not a root, f(r) will be equal to the remainder of the division.

the synthetic division is shown below:

the first part shows the steps in the synthetic division.
the second part shows the resulting equation after the division has been performed.
the third part shows that the remainder must be equal to 0 if r is a root of the equation which occurs if (x-r) is a factor of the equation.

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