Tahmid Knox

2020-12-15

(a) To calculate: How to rewrite any polynomial without powers of x. (b) To calculate: The relation betweeb the new form to synthetix division and the remainder theorem.

avortarF

(a)A polynomial is written without powers of x by factoring out x from thenon-constant terms.Given:A fourth-degree polynomial in x such as $3{x}^{4}+5{x}^{3}+4{x}^{2}+3x+1$contains all of the powers of x from the first through the fourth.However, any polynomial can be written without powers of x. Evaluatinga polynomial without powers of x (Horner's method) is somewhat easierthan evaluating a polynomial with powers.Calculation:A polynomial is written in the form,$P\left(x\right)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+{a}_{n-2}{x}^{n-2}+.....+{a}_{0},$ Which is arranged from thehighest to the lowest degree of x.Now, x is factored out from the non-constant terms of the polynomial,and proceeding in this manner, a polynomial can be written as withoutthe power of x.Therefore, a polynomial is written without powers of x by factoring out x from the non-constant terms.(b)The new form uses addition and multiplication to evaluate a polynomial.Given:A fourth-degree polynomial in x such as $3{x}^{4}+5{x}^{3}+4{x}^{2}+3x+1$contains all of the powers of x from the first through the fourth.However, any polynomial can be written without powers of x. Evaluatinga polynomial without powers of x (Horner's method) is somewhat easierthan evaluating a polynomial with powers.Calculation:A polynomial is written in the form,$P\left(x\right)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+{a}_{n-2}{x}^{n-2}+.....+{a}_{0},$ Which is arranged from thehighest to the lowest degree of x.Now, the new form uses precisely addition and multiplication in thesame manner as that of the synthetic division to evaluate a polynomial.Therefore, the new form uses addition and multiplication to evaluate apolynomial.