47. a) If (x – 2) is a factor of 2x – 3x + k, find k. - b) When ax – 4x² + 5x – 3 is divided by (x + 2) and (x – 1), the remainders are equal. Find a.

Q: Show that (x- 3) is a factor of 2.x3 - 2x2 - 17x + 15. Hence express 2x³ – 2.x² – 17x + 15 in the…

A:

Q: 7 If ZA = ZB, find mZA. (2x + 5) (50 - 4x) A B.

A:

Q: Prove that for integers y=1 the equation 3h-n (2x₁ + x₂ + 4x₂) +3² (~1(2x₁+x₂+.+xn-1)+...+3^-1 (2²₁)…

A: We will use contradiction to prove it.

Q: Use the elimination method to find all soluti { % 4x - Зу 3-24 -7x + 6y = { By = 45

A: Given,             The system of equations are                4x-3y=-24                -7x+6y=45

Q: a. Give the smallest extension of Q that contains all the zeros of x4 - 2x2 – 15.

A: We shall answer first question only as you have asked diff question without mentioning. For others…

Q: Show that x19 + x8+ 1 has multiple zeros in some extension of Z3.

A:

Q: 9. If mZSPR = (4x - 7)° and MZSQR = (2x + 3) °, find MZSPR.

A: Given that:   ∠SPR=(4x-7)°∠SQR=(2x+3)°

Q: Prove that there exists at least one solution to the equation 4x+ 3x +1 = 4x2.

A:

Q: Suppose X ÷ ( – 2) is 7x? – 2x – 6 with remainder 5. - - Then X %3D

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Q: (c) Express the quadratic form Q(x1,42. 43) = 6xj + 3x+3x-4xx2 - 21213 + 4xar1 to the sum of square,…

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Q: ) Express 8-x over Q[x] and over F2[x] as a product of irreducible ynomials.

A:

Q: 5. For how many different integers bis the polynomial x² + bx + 16 factorable over the integers?

A: From the given data

Q: (c) Express the quadratic form Q(x1,x2, x3) = 6x + 3x3 + 3x3 – 4x1x2 - 2x2x3+4x3x1 %3D to the sum of…

A: A detailed solution is given below.

Q: If the result of a polynomial division is 2x 3 − 4x 2 − 5x + 23 x + 2 , make at least three…

A:

Q: Q1. Find the general solution for the non-homogen 1st OLDE using undetermined coefficient method.…

A: Given Y'=-3-456Y+5-6ex

Q: 1. Determine the number on y=-x+3 2x-3y -24 %3D

A: The red line represents y=-x+3 , and the blue line represents 2x-3y=-24

Q: 6- Show that the Polynomial has a Z4 [x] 2X+1 in multiplicative invers. in 24 [x].

A: Given

Q: the smallest possibl product of the two numbers x>= 0 and y<= 0, such that 2x-y=120 is:

A: Let x and y be two numbers such that  x≥0,y≤0, 2x-y=120y=2x-120

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?

A:

Q: What is the coefficient of (x^8)(y^12) in the expansion of (2x+3y)^20

A:

Q: (b) Show that factorial polynomial corresponding to P,- k³ – 4k* + 2k³ – k² + k + 10 is P, = k(5) +…

A: Given: The given polynomials is Pk=k5-4k4+2k3-k2+k+10, To prove Pk=k5+11k4-k3-9k2-k1+10

Q: Show that r3 + 3x² – 8 is irreducible over Q.

A: Here Q is a field and f(x)= x³+3x²-8 is of degree 3. Then by ROOT TEST we can say that f is…

Q: What happens to y=fx) when you replace y with -4y? O vertical stretch by a factor of 4 and a…

A: Vertical stretch: The graph of y=dfx is the vertical stretch of the graph of y=fx by a factor d,…

Q: 9. Find the quotient and the remainder of x³ + 2x² – 1 divided by 3x2 +1 in Z7[x].

A: Note that for Z_7  Any number is equal to the remainder when it is divided by 7.

Q: 4. Prove that (2x+3y)(4x²−6xy+9y²)+(2x−3y)(4x²+6xy+9y²) is 16x³ equivalent to 1 for x = 0.

A:

Q: 1. It is known that the polynomial p(x) = 2 – (x + 1) + x(x + 1) – 2æ(x + 1)(x – 1) interpolates the…

A: Solution by using Newton's forward difference interpolation method .

Q: Show, by using the discriminant, how many solutions 2x2-x-5=0  has. must show all work

A: The discriminant of a quadratic equation ax2+bx+c=0 is given by D=b2-4ac.  We have h quadratic…

Q: 1) ciated with the quadratic form )² – 5x,x, + 6(x,)2.

A: We have to solve the problem

Q: 1. Show that 2x2-K = - (2K-1D has rationgi roots for all rational values of K.

A:

Q: Express as a polynomial. (2u +3)(u - 4) ( 4u+u ) 2

A: Given: y=22u+3u-44u+u For expressing given expression as a polynomial, we simplify given expression…

Q: Using Taylor's theorem, formally represent 2x° +x* – 3x³ +x² – 5x +1 as x+1.

A:

Q: Show that If 2 is a root of  kx4 - 11x3 + kx2 + 13x + 2 Then K = 3

A: Given,          2 is a root of kx4-11x3+kx2+13x+2 then k=3.

Q: (2) Determine k such that kx – 4x+=0 has real and different solutions. 2

A:

Q: While reducing the fraction 200..0z 300...0y (x, y + 0) in which between 2 and a and between 3 and y…

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Q: 2) When a polynomial, ax-3x² - bx - 5 is divided by the binomial (x-3), the remainder is 10. When…

A:

Q: The remainder upon the division of x75 + 6 by x-3 in z, is: 1.

A: concept : when f(x) is divided by x-a then  Remainder is f(a).

Q: (2) Determine k such kx² – 4x +- = 0 that has real and different solutions.

A: Note: - Since you have asked multiple questions, we will solve the first question for you. If you…

Q: 4. a. Prove that 8s + 13 L-1 (s² + 4s – 5) 5(7et +9e-5t) by using completing the square method.

A: Since there are two questions posted, we will answer the first question 4.a). If you want the…

Q: 5. Show that: a. x* + 9x³ + 4x + 7 is O(x*) b. x? + 4x + 17 is O(x³)

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Q: 22. Find the quotient and the remainder for each of the followir x3-4x2+2x+5 1) x-2 4x3-2x2-3 2)…

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Q: what is the multiplicity of -1 in x^4-2x^3-3x^2+4x+4

A:

Q: Show that the given values for a and b are lower and upper bounds for the real zeros of the…

A: Solution:- Any given value is said to be upper bound for the zeros of the polynomial if all the…

Q: is a factor of 2 + 5z2 – 5x +1. - O I+1 O 2z +1 O +2 O 2x -1

A:

Q: 1. What must be multiplied to (x2 + 2x + 1) to get (x + x + x2 + 3x + 2)? С. -ха- х - 2 D. x2 - x +…

A: Since you have posted multiple questions in this question which are not interlinked so I have solved…

Q: (2) Determinek such kx - 4x+- =0 that has real and different solutions.

A:

Q: ues of p will 3x(4x+3)=-p have non-real root

A: Given Equation 3x4x+3=-p The value of p is found such that the equation has a non-real root.

Q: In ARST, mZR = (6x + 10)°, mZS = (2x + 5)°, and mZT = (3x – 11)°. Find mZR. -

A: The sum of three angles of a triangle is equal to 180°. Equate the sum of the given three angles to…

Q: 1. Show that x + 4 is a factor of x* – x3 + 3x² – 3x + 356. -

A:

Q: 4. Prove that f(r) = 4x³ - 8² + 2x + 1 has three real zeros. %3D

A: