Polynomial equation with real coefficients that has the roots
is
Given:
Formula Used:
Calculation:
If the polynomial has real coefficients, then it’s imaginary roots occur in conjugate pairs. So, a polynomial with the given root
must have another root as
Since each root of the equation corresponds to a factor of the polynomial, also, the roots indicate zeros of that polynomial, thus, the polynomial equation is written as,
Further use arithmetic rule,
Now, the polynomial equation is:
Hence, the polynomial equation of given roots
DISCOVER: Nested Form of a Polynomial Expand Q to prove that the polynomials P and Q ae 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(2) 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 ofa polynomial using synthetic division?
(b) Express the polynomial 7x*-3x+1 as a linear combination of
Legendre's polynomials.