Question

QUESTION FOR
MATHS EXPERTS

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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).

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Answer 1

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||✪✪ QUESTION ✪✪||

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).

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|| ✰✰ ANSWER ✰✰ ||

Given,

• p(x) = x³ - 3x² + x + 2
• q(x) = x - 2
• r(x) = -2x + 4

By division algorithm, we know that

Dividend = Divisor × Quotient + Remainder

p(x) = q(x) × g(x) + r(x)

=> x³-3x²+x+2 = (x-2) × g(x) + (-2x+4)

=> x³-3x²+x+2+2x-4 = (x-2) × g(x)

=> g(x) = x³-3x²+3x-2 / x-2

On dividing x³-3x²+3x-2 by x-2,

We get

$\boxed{\begin{array}{l | c | r}\sf x-2&\sf x^3-3x^2+3x-2&\sf x^2-x+1\\&\sf \:\:\:\:\:\:\:\:x^3-2x^2\qquad\qquad\qquad\\&\:\:\:\:\:\:\:\:\:\:( - )\:\:\:\:( + )\qquad\:\:\:\qquad\qquad\\&\rule{90}{0.8}\quad\:\:\:\\&\sf\:\:\:\:\:\:\:\:\:\:\:\:-x^2+3x-2\qquad\\ &\sf\:\:\:-x^2+2x\qquad\!\!\\ & \:\:\:\:\:( + )\qquad( - )\qquad\\&\quad\rule{85}{0.8}\\&\qquad\qquad\sf\:x-2\\ &\sf\qquad\qquad\sf\:x-2\\&\qquad\qquad(-)\quad(+)\:\\&\qquad\quad\rule{65}{0.8}\\ &\sf\qquaqd\qquad\:\:\:\:\:\:\:\:\:0\end{array}}$

First term of g(x) = x³/x = x²

Second term of g(x) = -x²/x = -x

Third term of g(x) = x/x = 1

Hence, g(x) = x² - x + 1

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Answer 2

Answer:

x²-x+1 is the correct answer.

☸Given☸

• Dividend= x³-3x²+x+2
• Quotient= x-2
• Remainder= -2x+4

✏To Find:

• Divisor or g(x)

Remainder Theorem:

Dividend= Divisor × Quotient+Remainder

☯Solution☯

By using Remainder Theorem,

x³-3x²+x+2= g(x)×(x-2)+(-2x+4)

x³-3x²+x+2= g(x)×(x-2)-2x+4

x³-3x²+x+2+2x-4= g(x)×(x-2)

x³-3x²+3x-2= g(x)×(x-2)

$\sf{\dfrac{x^{3}-3x^{2}+3x-2}{x-2}=g(x)}$

Further calculation is in the attachment

After calculation, we get

$\longrightarrow\sf{\pink{g(x)=x^{2} -x+1}}$

Hence, the value of g(x) is x²-x+1.