If the polynomials fi and f2 are not divisible by the polynomial g, can théir sum fi + f2 be divisible by g? Explain in detail.

Q: . Do the polynomials x? + 2, 2x? – x + 1,x + 2, and x² + x + 4 span P2?

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Q: Explain why all Odd degree Polynomials with Real Coefficients have a Real Root.

A: We have to explain that odd degree polynomials with real coefficient have a real root. We know ,…

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Q: Express the polynomial a+ 3 +a2 + 2r as a product of its irreducible factors in Z,lr).

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Q: Express f(x)= 4+ 6 x² + 7 x + 2 in terms of Legendre Polynomials.

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Q: Consider two polynomials P (x)=4x-5 and Q(x)=3x2+7x-2 (a.) P(x) times Q(x)= (b.) P(-5)+Q(-3)=…

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Q: Express f (x) = x + 3x - x2 + 5x – 2 in terms of Legendre polynomials.

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Q: A simplified polynomial that has exactly one term is called a/an ............ ?

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Q: Do the polynomials x? + 2,2x2 – x + 1,x + 2, and x² + x + 4 span P2?

A: Let u1 = x2+2, u2 = 2x2-x+1, u3 = x+2, and u4 = x2+x+4. Notice that u1 + u3 = u4. So, u1, u2, u3,…

Q: A simplified polynomial that has two terms is called a/an ............ ?

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Q: Find the polynomial to represent (3x-7)2. O 9r2 - 21x + 49 O 9x2 + 21x + 49 9z2 – 42x + 49 9x2 + 42x…

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Q: What is a polynomial in x?

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Q: Prove the factor theorem.

A: Consider a polynomial px of degree n where n≥1. Then, the factor theorem states that x-c is a factor…

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Q: 3.4. Do the polynomials x3 + 2x, x² + x + 1, x³ + 5 generate (span) P3? Justify your answer.

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Q: How do you write polynomial funtions in standard form when the zeros are 5, 3i, and -3?

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Q: A local city wants to compare the number of those with a car and those without a car, represented by…

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Q: determine if the statement is true or false. If a statement is false, explain whyThe product of two…

A: Statement: The product of two polynomials each of degree 4 will be less than degree 8.

Q: 1. How do you know that the given polynomial is divisible or not? 2. How will you prove that the…

A: Given: A polynomial f(x) To check: 1) Whether the polynomial is divisible or not.

Q: Use long division to divide the following polynomials: (3ct + a3-#+1) 은 (22-2x + 1) O 3a2-5x +12 +…

A: Answer

Q: 3. Is the polynomial f(x) = 2x° + 30r – 150 € Q(r] reducible or irreducible over Q? In either case,…

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Q: 1. Compute the resultant of x – 3x* – 2r + 3x + 7x + 6 and x* +² + 1. Do polynomials have a common…

A: Given: To calculate the resultant of .

Q: Find a, beR such that the polynomial f(x) = 6x4 – 7x3 + ax? + 3x + 2 is divisible by g(x) = x2 – x +…

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Q: Prove the polynomial identity for the cube of a binomial representing a difference: (x-y)3=x³-3x?y +…

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Q: (d) Are there monic FSP polynomials (of some degree) that have real number coefficients, but some of…

A: Solution of part (d): Suppose there is a monic FSP polynomials of degree k that have real number…

Q: 6. Which polynomials have 4 roots? Select all that apply. 3x* – 81 = 0 4x + 4x + 4 = 0 2x4 + 3x? +…

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Q: The standard notation of the factorial polynomial x(4) - x(3) - 2

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Q: Which of the following polynomials is reducible over Q: x3 +2 x4 + 3 x5 + 1 x2 + 3 I thought it…

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Q: Multiply the polynomials (x-1)(x* + x+x² + x + 1) using a table. Generalize the pattern that emerges…

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Q: Determine whether the following polynomials span P2. P1 = 4- x+ 11x2, p2 = 7+ x, p3 = 29 + x + 22x?,…

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Q: Form the polynomial function that has the following real ZEROS: -6, -5, 3, 4 Section 3.1 O a4 + 4x -…

A: Explained below Proceed as shown...

Q: A third degree polynomial passes through the points (0,-3),(,0),(2,9), (3,30), (4,69) and (5, 132),…

A: Using Newton's forward difference method:f(x) = f(x0) + (r)∆y + (r)r-12!∆2y +  (r)r-1(r-2)3!∆3y…

Q: Determine whether the following polynomials span P2. P1= 3-x+ 6x2, p2 = 4 + x, p3 = 22 + 2x+ 12x²,…

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Q: 9) How can we classify polynomials? Classify these polynomials based on number of terms. a) x*-8x+2…

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Q: Determine whether the following polynomials span P2. P1 - 2-x+ 6x?, p2 = 7 + x, P3 = 32 + 2x + 12x²,…

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Q: Let R[:] be the set of all expressions a = ao+ajx+a2x² + …=La;x i=0 where a¡ e R for all nonnegative…

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Q: Sibelle manually counts maclaurin polynomials to sqrt(1+x) of order 1, ... 4. How?

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Q: Do the polynomials x² + 2, 2x² – x + 1,x + 2, and x² + x + 4 span P2?

A: Solve the problem as shown below see if they are linear combination of each other and then verify,

Q: division (X^3 -5x^2 -2x + 14) ÷ ( x -1)

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Q: Find the polynomials whose zeroes are (i)3,-4 and (ii)3, 3.

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Q: It can be proven that the expression (a +b) a - ab +b-) is equivalent to which of the following…

A: Given,            a + b a2 -ab + b2

Q: 2a] aS a product Of irreducible polynomials.

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Q: Show the polynomials given below have no integer roots. (a.) f(x) = x + 6x4 + 12x³ + 18x – 30, %3D…

A: To show that the following polynomials have no integer roots: a. fx=x7+6x4+12x3+18x-30 b.…

Q: Determine if the statement is true or false. If a statement is false, explain why?The sum of two…

A: To find:  If the statement, the sum of two polynomials each of degree 5 will be degree 5, is true or…