1. Show that 2x2-K = - (2K-1D has rationgi roots for all rational values of K.

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Q: Show that there is a root of the equation x3 - x - 1 = 0 between 1 and 2.

A: Consider the given equation. x3-x-1=0 and, fx=x3-x-1 Put x=1 in the function. f1=13-1-1=-1

Q: Show that if ya – ay = 1 has no rational solution for positive a such that a is a natural number and…

A: The given equation is ya-ay=1,a≠2 this equation is not valid for a = 1 although it is given for…

Q: The polynomial of degree 5, P(r) has leading coefficient 1, has roots of multiplicity 2 at z = 4 and…

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Q: Let 0 be a root of the polynomial x³ + 3x + 7. Prove that 0 is prime in Q(0).

A: We will use the basic knowledge of polynomials and rings to answer this question properly, in…

Q: 12 The non-permissible value(s) for the rational expressions 4 is (are) x2-4

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Q: Give an expression for f(x) with leading coefficient –5, of degree 3, and having roots ¤ : -4 , x =…

A: from the given data: i^2= -1 Substituting in the required place and finding the polynomial

Q: The polynomial of degree 5, P(x), has leading coefficient 1, has roots of multiplic 2 at x = 2 and x…

A: roots : x=2,2,0,0, -5 Factors are given by:x-2,x-2,x-0.x-0,x+5hence P(x)=Product of factors…

Q: The polynomial of degree 5, P(x), has leading coefficient 1, has roots of multiplicity 2 at r = 4…

A: Given that the polynomial is of order 5. Root of multiplicity is 2 at x=4, x=0. Root of multiplicity…

Q: 3. Find the real roots of the following rational equations. – 7x a. 9x + 11 -12=1 9x +11 X-1 x+2…

A: Given, (a) -7x9x+11 -12 = 1x  (b) x-1x+2 =3x+85x-1   We have to find the real roots of given…

Q: 6. Show that P(z) =r³ +x + 1 does not have any rational zeros. (Hint: What are the only…

A: Px=x3+x+1 Here the leading coefficient is 1 and constant term is 1. Now the factors of leading…

Q: 14. The condition for the quadratic cquation x2 + 2kx + (k + 6) = 0 to have two uncqual real roots…

A: When a quadratic equation have two real unequal real roots then the value of discriminate is greater…

Q: Q1: Find an approximation “ x2 “ to one of the zeroes of p(x) = x³ -x -1 using Newton-Raphson and…

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Q: If the result of a polynomial division is 2x 3 − 4x 2 − 5x + 23 x + 2 , make at least three…

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Q: The polynomial of degree 5, P(x) has leading coefficient 1, has roots of multiplicity 2 at x = 3 and…

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Q: Show that there is no rational number r satisfying 2r = 3.

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Q: A polynomial of degree n  1 has exactly zeros ifa zero of multiplicity m is counted m times

A: A polynomial of degree n≥1 has exactly  ___________  zeros if a zero of multiplicity m is counted m…

Q: 16.J. Show that every polynomial of odd degree and real coefficients has a real root. Show that the…

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Q: 11-31. Apply the procedures your team developed for adding fractions to add these rational…

A: First take the LCM as the common denominator for both fractions.

Q: Use a direct proof: If x and y are rational, then 3x+2y is also rational.

A: In this question, concept of Direct proof is applied. Direct Proof A direct proof is a series of…

Q: 17. Find all roots of the polynomial P(x) = x* – x³ – x² - x– 2.

A: This question is about nth root of polynomial

Q: 5x^2−3x−8 has roots at x=

A: Consider the quadratic equation To find - roots

Q: Find a polynomial f (x) of degree 5 that has the following zeros. 3, -4 (multiplicity 2), 7, -2…

A: Find a polynomial F(x) of degree 5that has the following Zeros3,-4 (multiplicity 2) , 7,-2

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: Show that ax2 + 2bx + c dx | (Ax² + 2Bx + C)² will be a rational function of x if and only if one or…

A: The given integral is ∫ax2+2bx+c(Ax2+2Bx+C)2dx Let us assume that the integral is a rational…

Q: #7 using Descartes's rule of sign

A: Part (a) The polynomial f(x) = x3 +3x +7 has a degree 3, so, it has 3 roots. For positive real…

Q: Suppose a polynomial of degree 4 with rational coefficients has the given numbers as zeros. Find the…

A: Surd zeros are always in pairs. That is a+b is zero then a-b is also a zero of the polynomial.…

Q: 8. Find all the roots of f(x) = x' - 4x' +6x² – 4x+5 if 2– i is one root.

A: We will find all the remaining roots as following.

Q: The polynomial of degree 5, P(x), has leading coefficient 1, has roots of multiplicity 2 at x=4 and…

A: Topic - polynomials

Q: In the open interval (0, 1), the polynomial p(x) = x4 - 4x3 + 2 has %3D 1. one real root three real…

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Q: 4. Find the real roots of the following rational equations. x+7 X-3 %3D x-5 x+8 X-11 2х-22 618

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Q: The polynomial of degree 5, P(x), has leading coefficient 1, has roots of multiplicity 2 at x = 3…

A: We want to find required polynomial.

Q: Use Steffensen's method to find, to an accuracy of 10-6, the root of x −2¯* =0 that lies in [0, 1].

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Q: Find the second-degree polynomial P(x) such that and P(0) = 1, P'(0) = 0, P'(2) = 4 and P(x) x³ (x –…

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Q: 15. What are all the possible rational roots of f(x) = 6x* – x² – 4x² – x – 2? ±1. ±2. ±3, ±6, ±늘,…

A: Rational root problem......

Q: The possible rational zeros of P(x) = x ^ 3 + 2x ^ 2 - 5x - 6 are _____, ______, __

A: we will use hit and trial method

Q: The polynomial f (x) leaves the remainder x + 2 at the division with x ^ 2 + 2x-3 and leaves the…

A: Use the remainder theorem for the polynomials to determine the remainder when f(x) is divided by…

Q: Find a formula for the polynomial P(x) with • degree 10 • leading coefficient 1 • root of…

A: Solution is given below:

Q: 5.5 (a) Determine the roots of f(x)= -12 – 21x+ 18x? – 2.75x3 graphically. In addition, determine…

A: Solution 5.5: f(x)=-2.75x3+18x2-21x-12 The graph of the function between xl=-1 and xu=0 Since the…

Q: Is the function in f(x)=(9√x+11)/(9x+11) a rational function? If so, write it in the form p(x)/q(x),…

A: Given function is,

Q: Generalize Theorem 1.3.9 by proving that every rational solution of a polyno- mial equation x" +…

A: Give polynomial is xn+an-1xn-1+…+a1x+a0=0, where all the coefficient's ak are integers.  Let p,q are…

Q: the polynomial of degree 4, p(x) has leading coefficient 1 had a zero of multiplicity 2 at x=2, a…

A: We are given that the polynomial p(x) has a zero of multiplicity 2 at x=2, i.e.; the polynomial has…

Q: The polynomial of degree 5, P(x) has leading coefficient 1, has roots of multiplicity 2 at a = 2 and…

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Q: Prove that the equation a3 – 4.x + 2 = 0 has three irrational solutions. Hin %3D but just show that…

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Q: Consider the following polynomial. H(x) = x³ + x² – 10x – 10 - Step 1 of 2: Use the Rational Zero…

A: Rational zero theorem :-  Let P = anXn +an-1Xn-1........+a0 =0 be a polynomial with an.......a0…

Q: Prove that the equation x- 15x +k = 0 has at most one root in the interval [-2, 2]. Hint: You'll…

A: Dear friend,  if you have any other query about this problem, then please tell me about your query I…

Q: 9х + 18 Consider the rational function f(x) 2х — 11 a. What monomial expression best estimates t…

A: The given rational function is:fx=9x+182x-11

Q: x to the third minus 512 over x-8 and must be solved using synthetic division.

A: First of all, consider the dividend and the divisor.

Q: The polynomial of degree 5, P(r), has leading coefficient 1, has roots of multiplicity 2 at z = 1…

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: