Question

Find the polynomial equation whose roots are the translation of the roots of the equation \[{x^5} - 4{x^4} + 3{x^2} - 4x + 6 = 0\] by \[ - 3\]

Answer 1

Hint: A polynomial equation is an equation that has multiple terms made up of numbers and variables. The highest exponent of the variable in the polynomial is called the degree of the polynomial. Polynomials can have different exponents. The degree tells us how many roots can be found in a polynomial equation. The roots of the polynomial equation are the values of \[x\] when \[y = 0\]. Here the translate word is used in the above question which means (increment). To translate the roots we will replace ‘\[x\]’ in polynomials by { \[x + \] value by which roots are translated}. Then what we get is a translational of the root of the given polynomial equation.

Complete step-by-step answer:
Given polynomial equation is
\[{x^5} - 4{x^4} + 3{x^2} - 4x + 6 = 0\]
Translate the roots of the equation by \[ - 3\]
Then, every root of equation will increase by \[ - 3\] in order to get new roots or polynomial equation whose roots are to be translate
So, we replace \[x\] with \[x + \] given translate value
i.e. \[x \to x + ( - 3)\]
\[x \to x - 3\]
Put this \[x - 3\] in given polynomial
\[{(x - 3)^5} - 4{(x - 3)^4} + 3{(x - 3)^2} - 4(x - 3) + 6 = 0\]
Which is required polynomial equation whose roots are to be translate by \[ - 3\]

Note: The number of zeroes of a polynomial is equal to the degree of the polynomial, and there is a well-defined mathematical relationship between the zeroes and the coefficients. ... Here, the real number k is said to be a zero of the polynomial of \[p\left( x \right),\]if \[p\left( k \right) = 0\].
There are various methods to find the zeros of the polynomial; we can do this with classical mathematics. We can also use calculus i.e. differentiation. All the method will lead to the same answer