a) Show that x = 1+2i is a root of p(x) where i = surds(-1). b) By using the fact that p(x) is a polynomial with real coefficients, show by contradiction that  p(x) will have another complex root.

Q: The polynomial f(x)=x^4+6x^3+ax^2-54x+c has four real roots r1, r2, r3, r4 such that r1+r2 = 0 and…

A: The polynomial f(x)=x^4+6x^3+ax^2-54x+c has four real roots r1, r2, r3, r4 such that r1+r2 = 0 and…

Q: Let a be a root of the polynomial x3+x+3, which is irreducible over Q. The polynomial x2-2 is also…

A: Given: a is a root of the polynomial x3+x+3, which is irreducible over ℚ. The polynomial x2-2 is…

Q: Form a polynomial f(x) with real coefficients having the given degree and zero  Degree 4; -5+3i; -1…

A: It is known that if a+bi is a zero of polynomial with real coefficients then a-bi also a zero. Given…

Q: A polynomial f (x) with real coefficients and leading coefficient 1 has the given zero(s) and…

A: The given zeros of a polynomial is 2 , -2-5i

Q: 5. What values should be assigned to coefficients a1, a2, and az, so that the quadrature formula…

A:

Q: 5. Find a polynomial of degree 4 with the zeros 3i and-v6j

A: Explanation of the solution is given below...

Q: Prove that if a > 0 and n is any positive integer, then the polynomial function p(x) = x2n+1+ ax…

A: Roots of the polynomial are values are when put for variable gives 0 value for the polynomial. At…

Q: Find the number of ideals in Cx,y that contains both the polynomials y – (x – 1)³ +1 and p(x) in…

A: We have to find the number of ideals in ℂx,y that contains both the polynomials y-x-13+1 and px in…

Q: Express the polynomial x3 + 2x+3 EZ-[x]as a product of irreducible polynomials of 75[x].

A:

Q: A polynomial f (x) with real coefficients and leading coefficient 1 has the given zero(s) and…

A: Given- Consider the roots 3+5i and -1-i. To find- A polynomial f (x) with real coefficients and…

Q: A polynomial f (x) with real coefficients and leading coefficient 1 has the given zero(s) and…

A: Given: A polynomial f (x) with real coefficients and leading coefficient 1 has the given zero(s) and…

Q: the given degree and zeros. Degree​ 4; ​ zeros: −4−2i;  −1 multiplicity 2

A:

Q: Prove Viete's Formulas: The quadratic polynomial with ∝ and β as roots is     x2 +bx+ c, where b =…

A: Viete's formula relates the coefficient of the function with the roots of the function. Sum of the…

Q: Find a polynomial with integer coefficients that satisfies the given conditions. Q has degree 3, and…

A: Let Q (x) be a polynomial that has degree 3, and zeros 0 and i. We know that if a+ib is zero of a…

Q: Ex.5 - Prove that the polynomial 2ª + z + 1 has exactly two zeros in the right half-plane.

A: Solution: We will be using the method of Routh array For a polynomial Qz=anzn+an-1zn-1+...+a1s+a0…

Q: find a formula for the polynomial P(x) with degree 3 leading coefficient 2 zeros at -5, -3, and 3

A:

Q: 9. Find a fourth-degree polynomial with integer coefficients that has zeros 3i and –1, with -1 a…

A: 3i and -1, with -1 a zero of multiplicity 2

Q: 2.33 Let p(r) = x4 + x³ -- 2x2 + x + 1. Prove that the polynomial equation p(x) = 0 has a root in…

A:

Q: 1 is a zero of the polynomial P(x) = x³ – 3x* + 7x³ – 13x² + 12x – 4 of multiplicity (а) 3 - - (b) 5…

A: 1 is the zero of given polynomial.

Q: polynomial f(x) of degree n 1, the fundamental theorem of algebra guarantees at least complex zero.

A: The Fundamental Theorem of Algebra tells us that every polynomial function has at least one complex…

Q: 7. The fourth-degree polynomial f(x) = 230x* + 18x + 9x? – 221x -9 has two real zeros, one in [-1,0]…

A: As per the guideline, I am doing the 1st method only. Kindly repost the others separately.   Secant…

Q: Find a formula for the polynomial P(x) with degree 3  leading coefficient 1 zeros at 8, -6, and 7…

A:

Q: The fourth-degree polynomial f (x) = 230xt + 18x³ + 9x? – 221x – 9 has two real zeros, one in [-1,…

A: The given problem is to find the two real roots of the fourth degree polynomial at accuracy 10-6, we…

Q: .Find a monic polynomial f(x) of least degree over C that has the given numbers as zeros, and a…

A: Monic Polynomial: "Mono" stands for "Single" Therefore it is a type of polynomial having single…

Q: 6. True or False A polynomial function of degree 4 with real coefficients could have -3, 2 + i,2 -…

A: True or False A polynomial function of degree 4 with real coefficients could have -3, 2+i,2 - i, and…

Q: 1- f(x) = x³ + 2x +1 has 3 roots in Z6- a) True b) False ) a) True

A:

Q: Find a polynomial P(x) of degree 3 that has integer coefficients and zeros 1/2 and 3 - i.

A: 3-i+32+1=-i2+1=-1+1=0 So 3-i  is a zero of " x+32+1 ". . . . . . (1). 2·12-1=1-1=0 So 12 is a zero…

Q: 4. Consider the polynomial f(x) = x² + 1. (a) Show that f(x) is irreducible in (Z/3Z) [x]. (b) Show…

A:

Q: Suppose the coefficients of the cubic polynomial p(x) = a +bx+cx^2+dx^3 satisfy the equation:…

A:

Q: Find the roots of the polynomial f(x) = a + (2 – 3i)a* – (2+ 6i)æ³ - (6+ 2i)æ² – (3 – 2i)æ + i in C…

A:

Q: 2. Find a polynomial with integer coefficients that has degree 3 and zeros 3, 2i, -2i.

A:

Q: Does there exist a polynomial of positive degree in Z6[x] that is a unit?

A:

Q: A quartic polynomial P(x) has rational coefficients. If √2 and 1 + i are roots of P(x) = 0, what are…

A: The detailed solution is as follows below:

Q: Find p so that  x2 - 6x + p = 0  has one root (i.e. is a perfect square)

A: We will solve it by two methods Shortcut method to find p for the equation given is x2 + b x + c  is…

Q: Form a polynomial f(x) with real coefficients having the given degree and zeros  Degree 4; zeros;…

A:

Q: 2-1n 7LX1, Show that the distinct polynomial s x+x and x+X determine the Sange function From Zz bo…

A: We have to show that,  the distinct polynomials x4+x  and x2+x in ℤ3x determine the same function…

Q: Find a formula for the polynomial P(x) with degree 3 • real coefficients -D1 y-intercept at (0, 20)…

A:

Q: (d) The cubic Legendre polynomial is P2(r) =(5x – 3r). Prove that it is orthog- onal (over [-1, 1)…

A:

Q: 6. Does there exist a polynomial of degree 4 with integer coefficients that has zeroes i, 2i, 3i and…

A: Complex root must be in conjugate pair

Q: Find a monic polynomial f(x) of least degree over C that has the given numbers as zeros, and a monic…

A: Given :-   The zeros of the polynomial 3i , 1+2i. The complex roots occur in pair, so conjugate of…

Q: A polynomial f(x) with real coefficients and leading coefficient 1 has the given zero and degree.…

A: Given :- zero :- 5+9i and degree =2 To find:- A polynomial f(x) with real coefficients and leading…

Q: Find the cubic polynomial function f(x) with real coefficient that has 2, and 1-i as zero and f(1)=3

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: Find a polynomial of least degree with only real coefficients and having the given zeros. 2+i, 4 A.…

A: Here we have to find a polynomial of least degree with one real coefficients and having the given…

Q: B. Complete the table below: Polynomial Equatlons Degree Real Roots of an Equation Number of Real…

A: 4.x+32x-54x+25=0Degree=11Real roots =-3,5 and -2Number of real roots =11…

Q: Find all the roots of z4 + 1= 0 and use them to show that z4 +1=(z² – /2z +1)(z² + v2z + 1) -

A:

Q: Find a formula for the polynomial P(x) with degree 3 leading coefficient 1 zeros at 8, 5, and 0…

A: polynomial P(x) with degree 3 leading coefficient 1 zeros at 8, 5, and 0

Q: Find a polynomial of degree 4 with integer coefficients that has -2 as its only real zero and (1/3)i…

A: A polynomial of degree 4 with integer coefficients that has -2 as its only real zero. Which implies…

Q: Find a polynomial P(x) with rational coefficients that has degree 5 and zeros, -2, 1+2i, and -1-2i…

A:

Q: A polynomial f (x) with real coefficients and leading coefficient 1 has the given zero(s) and…

A: Given Zeroes= 4+3i, -2+i therefore conjugate of above zeroes are also zeroes of f(x) Therefore 4-3i,…

Q: find a formula for the polynomial P(x) with degree 3 leading coefficient 1 zeros at -6, 4, and 7…

A:

Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.