Question

Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in
each of the following :
(i) p(x) = x³ – 3x² + 5x – 3, g(x) = x² – 2

(ii) p(x) = x⁴ – 3x² + 4x + 5, g(x) = x² + 1 – x

(iii) p(x) = x⁴ – 5x + 6, g(x) = 2 – x²​

Answer 1

Step-by-step explanation:

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${\bigstar\Huge{\boxed{\tt {\color{red}{Question}}}}}\bigstar$

$\hookrightarrow$ Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in

each of the following :

(i) p(x) = x³ – 3x² + 5x – 3, g(x) = x² – 2.

(ii) p(x) = x⁴ – 3x² + 4x + 5, g(x) = x² + 1 – x.

(iii) p(x) = x⁴ – 5x + 6, g(x) = 2 – x².

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${\bigstar\Huge{\boxed{\tt {\color{red}{Answer}}}}}\bigstar$

$\hookrightarrow$ 1). p(x) = x³ – 3x² + 5x – 3, g(x) = x² – 2.

$\hookrightarrow$ ● Quotient = x - 3.

$\hookrightarrow$ ● Remainder = 7x - 9.

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$\hookrightarrow$ 2). p(x) = x⁴ – 3x² + 4x + 5, g(x) = x² + 1 – x.

$\hookrightarrow$ ● Quotient = x² + x - 3.

$\hookrightarrow$ ● Remainder = 8.

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$\hookrightarrow$ 3). p(x) = x⁴ – 5x + 6, g(x) = 2 – x².

$\hookrightarrow$ ● Quotient = -x² - 2.

$\hookrightarrow$ ● Remainder = -5x + 10.

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Answer 2

(i) p(x) = x³ – 3x² + 5x – 3, g(x) = x² – 2

Solution:-

$\begin{array}{c|c|c}\sf x^2 - 2&\sf x^3 - 3x^2+5x-3&\sf x+3\\&\sf \qquad x^3 \quad\quad \: \: -2x\qquad\qquad&\\&\dfrac{\qquad \: \: \: \: -\ \:\:\quad\qquad\:\:\:+}{\sf \qquad\qquad\qquad- 3x^2 + 7x - 3}\qquad\qquad\quad&\\&\sf - 3x^2 \qquad \:\:\:\: + 6&\\&\sf\dfrac{ + \qquad\qquad \:\:\:\:\:\: - }{\qquad\qquad 7x - 9}\end{array}$

Remainder = 7x - 9

Quotient = x - 3

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(ii) p(x) = x⁴ – 3x² + 4x + 5, g(x) = x² + 1 – x

Solution:-

$\begin{array}{c|c|c}\sf x^2 - x + 1&\sf x^4\qquad - 3x^2+4x + 5&\sf x^2 + x - 3\\&\sf \qquad x^4 - x^3 + x^2\qquad\qquad\qquad\quad&\\&\dfrac{\qquad-\quad + \quad-\qquad}{\sf \qquad\qquad\qquad x^3- 4x^2 + 4x + 5}\qquad\qquad\quad&\\&\sf x^3- x^2 + x \: \: \: \: \: \: \: &\\&\sf\dfrac{ + \quad + \quad - \qquad}{ - 3x^2 + 3x + 5}&\\&\sf - 3x^2 + 3x - 3&\\&\dfrac{ +\quad - \quad + }{\sf \qquad\qquad8}\end{array}$

Remainder = 8

Quotient = x² + x - 3

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(iii) p(x) = x⁴ – 5x + 6, g(x) = 2 – x²

$\begin{array}{c|c|c}\sf - x^2 + 2&\sf x^4 \qquad\qquad\:\:\:- 5x+6&\sf -x^2 - 2\\&\sf \qquad \: \: x^4 \quad\quad \: \: \: \: \: \: \: \: \: \: \: \: -2x^2\qquad\qquad&\\&\dfrac{ \: \: \: \: \: \: \: \: \: -\ \:\:\quad\qquad\:\:\ \: \: \: \: +}{\sf \qquad\qquad\qquad 2x^2- 5x +6 }\qquad\qquad\quad&\\&\sf \: \: \: \: \: 2x^2 \qquad + 4 &\\&\sf\dfrac{ - \qquad \: \: \: - }{ \qquad \qquad5x + 2}\end{array}$

Remainder = 5x + 2

Quotient = -x² - 2

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