Use the Division Algorithm to prove that (a) The square of any integer is either of the form 3k or 3k + 1. (b) 3m² - 1 is never a perfect square.

Q: Given that one of the zeroes of the cubic polynomial ax3+bx2+cx+d is zero , the product of the other…

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Q: Show that x2 + 2x8 + 1 does not have multiple zeros in any extension of Z3.

A: As per the question we are given a polynomial x2 + 2x8 + 1 and we have to show that there is no…

Q: Jesse received a total of $10,000 as loans from his cousin John. After six years, Jesse paid $8,000…

A: Given: Jesse received a total of $10,000 as loans from his cousin John. After six years, Jesse…

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

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Q: Create a cubic expression that when divided by (x-1) will have a remainder of 6

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Q: Write p(x) = 375 -7³+57²-1 in the form of nested multiplication. Compare the computational cost…

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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: 38 a Complete the square for x² – 8x + 25. b Hence find 2x +7 dr. x² - 8x + 25

A: To complete the given square and to evaluate the given integral.

Q: 3. (x+7) What pattem do you see with these three perfect squares? Double check your hypotheses by…

A: Given x+72

Q: = (x+ )². 12 2²2 +6x +. = (x+ 2² + 6x +9= (x+3) ².

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Q: EXERCISE 8 1. DON'T SIMPLIFY the following. ONLY INDICATE whether or not the result of the product…

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Q: Phythagoras was the one who formulated one of the earliest theorems known to ancient civilizations…

A: Yes the given statement is true  Because approx 2000years ago a theorem is granted named as ancient…

Q: 2. Which of the following binomials is a factor of the quadratic expression, three x-twenty-six x +…

A: Consider the quadratic expression: 3x2-26x+16

Q: Factor the perfect square trinomial 4x2 - 20x + 25.

A: we use following steps: Order to factorize ax2+bx+c we have to find number p and q such that p + q…

Q: Find a counterexample to show that each of the statements is false.

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Q: For each of the following, fill in the blank to create a perfect-square trinomial: (a) x2 − 6x + ___…

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Q: In the 1989, Journal of Number Theory paper, N. Tzanakis and B. de Wegner showed that the largest of…

A: We have to factories the equation  x3+3x2+2x-258,474,216=0

Q: 8. Express (3x - 4)(x + 7) - ² a as a trinomial in standard form.

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Q: Prove that (x+2) is a factor of 3x'+8x+x-6 and find the other factor.

A: we have to find factor of the polinomial

Q: For the remainder of the division of x-2x²+3kx+18 by x-6 to be equal to zero, k must be equal to?

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Q: Indicate the least integer n for which (3x6 + 2x2)(x3 + 3x) is O(xn).

A: Solution:-

Q: Which of the given quadratic forms in three variables are equivalent?

A: Four quadratic forms in three variables are given. Find out which of the given quadratic forms in…

Q: factor 3x^3+17x^2-54x+16 into linear factors, Give all factors in exact form.

A: Topic- algebraic factors

Q: the number of nonnegative integer solutions to (x₁ + x2 + x3 + x4 + X5) (X6 + X7 + x8 + x9 + x10) =…

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Q: (2.1) Factor completely, if a polynomial cannot be factored, say so: a) 6 m*n – 9 m³ n²– 3 m² n³ b)…

A: Topic = Algebra

Q: Prove that square of any integer leaves the remainder either 0 or 1 when divided by 4.

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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: 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…