College algebra

Question

Nested Form of a Polynomial Expand Q to prove that the polynomials P and Q are the same.

P(x)=3x^4-5x^3+x^2-3x+5

Q(x)=(((3x-5)x+1)x-3)x+5

Try to evaluate P(2) and Q(x) in your head, using the forms given. Which is easier? Now write the polynomial

R(x)=x^5-2x^4+3x^3-2x^2+3x+4 

in “nested” form, like the polynomial Q. Use the nested form to find R(3) in your head. Do you see how calculating with the nested form follows the same arithmetic steps as calculating the value of a polynomial using synthetic division? 

Answer 1

Q(x) = ((3x^2 - 5x + 1)x - 3)x + 5

        = (3x^3 - 5x^2 + x - 3)x + 5

        = 3x^4 - 5x^3 + x^2 - 3x + 5

        = P(x) 

 

R(x) = (x^4 - 2x^3 + 3x^2 - 2x + 3)x + 4

       = ((x^3 - 2x^2 + 3x - 2)x + 3)x + 4

       = (((x^2 - 2x + 3)x - 2)x + 3)x + 4

       = ((((x - 2)x + 3)x - 2)x + 3)x + 4