Let a0, a1, a2, a3, a4 be constant real numbers such thata0/5 + a1/4 + a2/3 + a3/2 + a4 = 0.Show that the polynomial P (x) = a0x^4 + a1x^3 + a2x^2 + a3x + a4 has at least one zero between0 and 1. (Hint: Considerf (x) = a0/5 x^5 + a1/4 x^4 + a2/3 x^3 + a3/2 x^2 + a4x.Show that Rolle’s theorem applies to f (x) on the interval [0, 1]. Deduce that P (x) has a 0 in[0, 1]).

Q: The polynomial x(x^2-4)(x^2+4) has___ real roots.

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Q: 16. Find a fourth degree polynomial that has the zeros 3, -2, i.

A: Given data: The first root of the polynomial is x1=3. The second root of the polynomial is x2=-2.…

Q: 1 ) Write f(x) = x^5 + x^3 +2x^2 -12x +8 as the product of linear factors and list all the zeros of…

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Q: List all the possible rational rooots  for the polynomial F(X)=2x^2 -9x+3

A: By quadratic formula for roots of quadratic equation

Q: The fundamental theorem of Algebra states that f(x)=x^7+6x^3-5  Can have at least 5 real solutions…

A: f(x)=x^7+6x^3-5  has highest exponent= 7  so degree of f(x) is 7

Q: Write g(x) = x² + x³ + 2x²+x+2 as a product of irreducible polynomials over 2.₂. за

A: Write as a product of irreducible polynomials over .

Q: Use Descartes’s Rule of Signs to determine the possiblenumber of positive and negative real zeros…

A: Given function is f(x)=3x^5-2x^4-2x^2+x-1Firstly,we count the number of changes in signs for the…

Q: Prove that 3x3+ 5x - 11 has a root between x=1 and x=2. Thanks!

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Q: Express as a polynomial.

A: Express as a polynomial.         9p4q3-6p2q4+5p3q23p2q2

Q: Q3) Write in factor form a 5h degree polynomial whose zeros are +2, -2 and 4 with multiplicity 3

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Q: 4. Write the polynomial in standard form. * 4g – g° + 3g² – 2 -2 +4g + 3g – g g - 3g +4g - 2 O A -g…

A: Standard Form of Polynomial states that the polynomials needs to be written with the exponents in…

Q: Find a polynomial p(x) = ao + a1x + aza² + azx³ of degree < 3 (here ao, a1, a2, a3 are real numbers)…

A: In this question , use the given values and solve the equations and find the polynomial.

Q: It is possible to write xk as a linear combination of Chebyshev's polynomials, for example %3D…

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Q: 7 Quadratic polynomial factors should be typed as (x^2+ax+b)(x^2+cx+d)( :) Factor f(x) = x¹-2x³-x² -…

A: Ok

Q: A polynomial which goes through the point (1,4) and has zeros at x=0,x=3 and x=-2

A: Given:A polynomial pass through the point (1,4) and has zeroes at x=0, x=3 and x=-2.

Q: 15f(n)=(2n+1)^(2)-(2n-1)^(2)-10 where is an integer. Prove that f(n) is never a multiple of 8

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