Here we use division of algorithm
Explain everything clearly read carefully and understand.
We know by division algorithm
A(x) = Q(x)B(x) + R(x) ,where 0 < or = R(x) <Q(x)
This can be written as [A(x)÷B(x)] = Q(x) + [R(x)÷B(x)]; where degree of R(x) is less than degree of B(x).
R(Xwhere the degree of R(x) is less than the degree of D(x) D(x) Use polynomial division...
* 5. Form a polynomial f(x) with real coefficients having the given degree and zeros. Degree: 3; zeros: -2 and 2i (a) f(x) = x + 2x² + 4x +8 (c) f(x) = x2 – 2x² + 4x – 8 (b) f(x)= x + 2x2 - 4x + 8 (d) f(x) = x3 – 2x2 - 4x - 8
Problem 10.13. Recal that a polynomial p over R is an expression of the form p(x) an"+an--+..+ar +ao where each aj E R and n E N. The largest integer j such that a/ 0 is the degree of p. We define the degree of the constant polynomial p0 to be -. (A polynomial over R defines a function p : R R.) (a) Define a relation on the set of polynomials by p if and only if p(0) (0)...
3) Write a polynomial f(x) that meets the given conditions. Answers may vary. 3) Degree 2 polynomial with zeros 212 and -222 A) S(x) = x2 + 472x+8 B) f(x) = x2-8 9 S(x) = x² + 8 D) S(x) = x2-11/2x+8 4) Degree 3 polynomial with zeros 6, 21, and -2i A) S(x) => x3 + 6x2 + 4x + 24 f(x)= x2 - 6x2 + 4x - 24 B) /(x) = x2 - 6x2 - 4x + 24...
Find the quotient Q(x) and remainder R(x) when the polynomial P(x) is divided by the polynomial D(x). P(x) = 4x5 + 9x4 − 5x3 + x2 + x − 25; D(x) = x4 + x3 − 4x − 5 Q(x) = R(x) = Use the Factor Theorem to show that x − c is a factor of P(x) for the given values of c. P(x) = 2x4 − 13x3 − 3x2 + 117x − 135; c = −3, c = 3...
Find the complete factored form of the polynomial f(x) that satisfies the given conditions. Degree 3, leading coefficient -5, zeros at 9,2-8 i and 2 +8 i. O A. f(x)= - 5(x- 9)(x2 - 4x +68) OB. f(x) = -5(x +9)(x - 2 - 8i)(x - 2 +81) O c. f(x) = -5(x -9)(x - 2 - 8 i)(x - 2 + 8 i) OD. f(x) = - 5(x +9)(x2 - 4x+68)
Let (r) 2 +ar+ ... +,r" be a polynomial of degree less than or equal to n and letto ......} be distinct points What is the value of from... Is? Explain Please select files) Select files)
Definition. The degree of a a polynomial is the exponent on the the highest power of x. Polynomial Degree 210 - 5.0 + 6 10 3.C - 1 13 Exercise 4. Scheinerman Exercise 35.12. Consider polynomials in x with rational coeffi- cients. a) Suppose p and q are polynomials. Write a careful definition of what it means for p to divide q (i.e. plq). Verify that (2.1 – 6(x3 – 3.x2 + 3x – 9) is true in your definition....
subring of the polynomial ring R{z] (i Show that R is a (ii) Let k be a fixed positive integer and Rrk be the set of all polynomials of degree less than or subring of Ra (iii) Find the quotient q(x) and remainder r(x) of the polynomial P\(x) 2x in Z11] equal to k. Is Rr]k a T52r43 -5 when divided by P2(x) = iv) List all the polynomials of degree 3 in Z2[r]. subring of the polynomial ring R{z]...
Consider the polynomial f(x) = x+ + 4x + 4x? a. What is the degree of this polynomial?_ b. What is the y intercept?_ c. What are the roots (zeros) of this polynomial? d. What is the end behavior of this polynomial ? e. Sketch this polynomial?
Let f(x) = do +212 + ... + and" be a polynomial of degree less than or equal to n, and let {x0, 21, ..., Un} be distinct points. What is the value of f (x0, X1, ... , Xn]? Explain.