Question

sol 4th polynomial questions​

Answer 1

On dividing x³-3x²+x+2 by a polynomial g(x),the quotient and remainder were x-2 and -2x+4 respectively.find g(x)

f(x) = x³ - 3x² + x + 2

q(x) = x - 2

let say

g(x) = ax² + bx + c

r = - 2x + 4

f(x)  = g(x)q(x) + r

=> x³ - 3x² + x + 2 = (ax² + bx + c)(x - 2) + (-2x + 4)

=> x³ - 3x² + x + 2 =  ax³ + x²(b - 2a) + x(c - 2b) -2c -2x + 4

=> x³ - 3x² + x + 2 =  ax³ + x²(b - 2a) + x(c - 2b - 2)  + (-2c  + 4)

Equating Power terms

a = 1

b - 2a = - 3  

=> b -2 = -3

=> b = -1

c - 2b - 2 = 1

=> c + 2 - 2 = 1  

=> c = 1

⇒ -2c  + 4 = 2

 => -2c = -2

=> c = 1

g(x) = x² - x + 1

(Ans): g(x) = x² - x + 1

Answer 2

$\huge\mathfrak{Polynomial:}$

An algebraic expression with the combination of different variables and constant terms where the power of each real number is a non negative integer is what we call as polynomial.

Examples include : (√3z² - 5z + 6), (x -2), etc,.

There are different types of polynomials. Some of these are :

» Linear polynomial, of degree or power 1

» Quadratic polynomial, of degree or power 2, etc.

$\huge\mathfrak{Solution:}$

Let the given polynomial = p(x)

quotient = q(x) and remainder = r(x)

From the Euclids division lemma,

Dividend = (Divisor)(Quotient) + Remainder

i.e., p(x) = [g(x)][q(x)] + r(x)

On substituting all the given values,

g(x) = x^2 - x + 1

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Therefore, required g(x) = x^2 - x + 1

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For the exact calculations, refer to the attachment.