In each of the following, a polynomial P(x) and a divisor d(x) are given. Divide to...
Use long division to find the quotient Q(x) and the remainder R(x) when P(x) is divided by d(x) and express P(x) in the form d(x) • Q(x) + R(x). P(x) = x3 + 5x? - 17x+172 d(x) = x + 9 P(x) = (x+9)( +
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...
A polynomial Px) and advisor obx) are given Use long division to find the quotient Qpx) and the remainder REX) when PIX) is divided by dix), and express P(x) in the form d x) (x) POS=X. 54 dX) EX-2 ON -21072 - 2x + 4)+72 O X-2302 - 2x 4) - 56 OC -230 +234 OD -232 2x+4)*72
I need help with these questions. 1. Answer the questions below. (a) For the polynomial P(x) = 3.r* - 2x2 + 2x – 4, identify the following. i. (2 points) The degree of P ii. (2 points) The constant term of P iii. (2 points) The leading coefficient of P iv. (4 points) The possible rational zeros of P (b) (5 points) Find the quotient and remainder of when Q(x) = * - 223 +22 - 3 is divided by...
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]...
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...
(i) Show that R is a subring of the polynomial ring Rx. | R{]4 (ii) Let k be a fixed positive integer and be the set of all polynomials of degree less than or equal to k. Is R[xk a subring of R[a]? 2r4+3x - 5 when it is (iii) Find the quotient q(x divided by P2(x) of the polynomial P1( and remainder r(x) - 2c + 1 in - (iv) List all the polynomials of degree 3 in Z...
SECTION 4.3 Polynomial Division; The Factor the polynomial function f(x). Then solve the equation f(x) = 0. 39, f(x) =x3 + 4x2 + x-6 40. fx) 5x - 2x 24 41, f(x) =x3-6x2 + 3x+10 42. f(x)-x3 + 2x2-13x + 10 43, f(x) = x3-x2-14x + 24 44.f(x) = x3-3x2 In Ex given. a): Fi b) C in gi - L 二 10x +24ー丁only, this one d) C gi ase 45' f(x) =x4-7x3 + 9x2 + 27x-54 plecs( 46, f(x)...
(i) Show that R is a subring of the polynomial ring Rx. | R{]4 (ii) Let k be a fixed positive integer and be the set of all polynomials of degree less than or equal to k. Is R[xk a subring of R[a]? 2r4+3x - 5 when it is (iii) Find the quotient q(x divided by P2(x) of the polynomial P1( and remainder r(x) - 2c + 1 in - (iv) List all the polynomials of degree 3 in Z...
Answer ALL questions. (20 marks) 1. Find the quotient and remainder by using long division. (6x + 4x* + 3) + (1 + 2x) 2. Given that P(x) = ax2 + 5x - 1 and Q(x) = bx + 4. If P(x). Q(x) = 3x2 + 17x2 + 19x - 4, find the values of a and b. 3. The function H(x) = x3 + ax? – bx + 5 gives a remainder of 11 when divided by x +...