Show that if p is prime and p ≥ 7, then there are always two quadratic residues of p that differ by 3.

Q: 4. Find a quadratic polynomial with roots -1 and 2. A. a2 a + 2-0 B. a2- a 2-0 C. a2 + a 2 0 D. a2 +…

A: We have to find the quadratic polynomial with roots -1 and 2

Q: B. Find the value of k that makes x2 + (k + 5)x + (5k + 1) a perfect square.

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Q: If m<1 = 6(3x+1)-2, m<2= 3(x+2)+9, and m<ABC= 5(3x+5)+6, find x and m<ABC.

A: Given,           ∠1=63x+1-2           ∠2=3x+2+9            ∠ABC=53x+5+6

Q: Show that the square of any integer is of the form 3m or 3m+1 but not of form 3m+2.

A: Proof :-

Q: 8. (S.15). Write P20 r2+2r-1 as a linear combination of the polynomials P21 r-r- 2, P22 2r2-r+3, P23…

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Q: 6) Suppose the cubic polynomial 8x³ + mx² + nx - 6 has both 2x + 3 and x - 1 as factors. Find m and…

A: Given polynomial equation:- 8x3+mx2+nx-6

Q: j) x(x+2)+

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Q: b) Suppose x # 2. Find dif dx In y = In (x - 2)² √x² +1

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Q: Find the polynomials connecting the points (0,1), (1,3), and (2,13) ​​by quadratic interpolation.

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Q: 1. Given that r + px+q and 4x +q have a factor (x-r) where p,q and r are non-zero. Show that 4p…

A: topic - quadratic equations

Q: Prove that (p – q)× (p+q)= 2(p×q)

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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: 8 Select the Quadratic Binomial. x+8 B) 3x² + 2x -8. c) x³ x³ - 3x² Dx+7

A: quadratic polynomails of type ax2+bc and ax2+c are called quadratic binomails

Q: Prove that 3n² - 1 is never a perfect square.

A: Perfect square: An integer n is said to be a perfect square if n=k.k for some integer k  We now…

Q: 2. Find the difference quotient of f(x) = 4x² – 7x + 5

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Q: The maximum value of the quadratic form 4x2 – 8xy + 5y² is Select one: 9+65 2 9-V65 2 9 + V65 9+V185…

A: The given expression is u =4x2-8xy+5y2 calculate partial derivatives ∂u∂x=8x-8y=0⇒x=y…

Q: What is the constant term of the polynomial 5xº - 4x® + 2x5 + x³ + x? D. 0 A. 5 В. 8 С. 1

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Q: Suppose the second column of B is all zeros.  What can you say about the second column of AB?

A: If the second column of B is all Zeros, then the second column of AB will also be all Zeros.

Q: Find all possible pairs of integers x, y so that 7x + 3y = 4. Then find which of these pairs makes…

A: Given:   7x+3y=4                        ........(1) We have to find all possible pairs of integers…

Q: Find integers x and y so that gcd(2021, 271) = 2021x + 271y. Find integers m and n so that gcd…

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Q: Indicate the least integer n for which (3x⁰+2x²)(x³ +3x) x+14x € 0(x¹).

A: Given: 3x6+2x2x3+3xx4+14x∈Oxn To find: Least integer n for which the given condition holds.

Q: 1. What is the remainder if the polynomial 3x* + 2x3 – 4x2 + x+4 is divided by (x-2)? С. 300 D. 143…

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Q: Find the square root of the product of (3+ j4) and (2+ j8)

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Q: 10) If a, b and c are the roots of the polynomialz + 13r- 23 and sum of the roots is 0, then find…

A: The detailed answer is given below Thank you

Q: 2n b) The polynomial, fW =2x" +mx-3, where m and n are real constants, %3D is divisible by (x+1) but…

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