6. Let ao, a1, a2, a3, a4 be constant real numbers such that a1 a2 a3 + + + + a4 = 0. 5 4 3 2 Show that the polynomial P(x) = ax² + a₁x³ + a₂x² +a3x+a4 has at least one zero between 0 and 1. (Hint: Consider a1 4 a2 5 a3 f(x) = 2x³ + 2/1x²¹ +23²x³² +22³x² + 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: 25. f(x) =x-16x A) -4, 0, 4 B) -8, 0, 8 C) -2, 1, 2 D) 2, 4, 8 26. g(x) = x* -x' -6 A) -V3,-V2, 2, 3…

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Q: 5. Find a polynomial of degree 3 whose zeros are 4, 1,-2 and y intercept 16.

A: Since given that the polynomial has degree 3 and its zeros are 4, 1, and -2.Then the polynomial can…

Q: 13. Find the equation of the third degree polynomial containing the points (1, -7), (0, 4),(3, -2)…

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Q: The generally polynomials are of the form Select one: a. b. O C. d. an" + an-x-1 + ... + aix+ ao…

A: Polynomial A polynomial is  an expression which includes variables, constants and exponents, that…

Q: e down a polynomial of degree exactly 4 that interpolates the four points: (-1,-5 1), (2, 1), (3,…

A: We have to find the polynomial equation from given points.

Q: 2. Find the polynomial the passes through the following points (0,0), (2, —2), (4,0)

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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: Let a0, a1, a2, a3, a4 be constant real numbers such thata0/5 + a1/4 + a2/3 + a3/2 + a4 = 0.Show…

A: Consider−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−f(x)=5a0​x5​+4a1​x4​+3a2​x3​+2a3​x2​+a4​x.with  input from…

Q: Which of the positive integers coming from the set{1, 2, 3, 4} are upper bounds for the zeros of the…

A: The given polynomial is Px=x4-4x2-3

Q: When the polynomial f(x) = 3x³-7x2+4 is divided by g(x)=(x-2), the result is of the form ax2-bx-c,…

A: Divide the polynomial f(x) with g(x).  The compare the divisor.

Q: Which polynomial function would have the end behaviour of as x→ too, Y→ co? a) f(x) = 7x³ - 8x* -…

A: Topic = Functions

Q: Formulate that the Lagrange interpolation polynomial that passes through points (0, 3), (1,4) and…

A: To show that the Lagrange interpolation polynomial that passes through points (0,-3),(1,4),(2,15) is…

Q: A third degree polynomial passes through the points (0,9), (1.14), (2,17), (3,24), (4,41), (5,74)…

A: The value of X1=∆y3=y4-y3=24-17⇒X1=7and value of ∆2y2=X2=∆y3-∆y2=7-3=4⇒X2=4

Q: rmine all the zeros of f(x) = 3x121 + 5x64 – 2x54 +x44 in Z7[x]. (Hint: Use Ferma |3D

A: In this question, the concept of the Fermat Little Theorem is applied. Fermat Little Theorem…

Q: In Exercises 7-12, find all zeros of the given polynomial. 7. x³ + x² - x + 2 8. 3x³ - 7x² + 8x - 2

A: As student asked to solve 7. So,  here we solve question 7 only.  Here we have to find zeros of…

Q: 10. Show that the polynomial p(x)=x²-3x³-x² - 4x-20 has at least two distinct roots (Hin use the…

A: We have given,   f(x) = x4 - x2 - 4x - 20

Q: 7hat is a polynomial function in standard form with zeroes 1,-2, 3, and -3? Ox+x - 11x2– 9x- 18…

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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: 3. Which of the following polynomial function has degree = 3 and leading coefficient -6? a)…

A: according to our guidelines we can answer only three subparts, or first question and rest can be…

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

A: Let's find x value.

Q: Sketch a higher degree polynomial with the following roots: -2, 3, -3, 6. f(x) 9,

A: It is given that the roots are -2, 3, -3 and 6. That is, x+2=0, x-3=0, x+3=0 and x-6=0. The…

Q: A real number is called algebraic if it is a root of an algebraic equation f(x) = 0, where f(xo) ao…

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Q: Create a polynomial function of the form: f(x) = a (bx + e)(cx +g)²(dx + h)³, with the following…

A: To create a polynomial function of the below form that satisfies all the given conditions.…

Q: Which of the following are NOT polynomials? I.  \sqrt(x-9) II.   x^8−14x+1 III.  1/2⋅x+0.149 IV.…

A:

Q: #24

A: Hi there, you had mentioned #24. That is why I have answered question number 24.

Q: If a and B are the zeros of the quadratic polynomial f(x) = ax? + bx + c,then evaluate: (19 a? B.…

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Q: 5. Use end behaviours, turning points, and zeros to determine which graph represents the polynomial…

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Q: ow that x+a is a factor of the polynomial p(x)= (x+a)ª+(x+c)ª-(a-

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Q: 11. Summarize the Intermediate Value Theorem for Polynomials in your own words.

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Q: Create a polynomial function of the form: f(x) = a (bx + e)(cx +g}*(dx + h)°, with the following…

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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.