Question

Give examples of polynomial p(x),g(x) and r(x), which satisfy the division algorithm and

deg r(x) = 0​

Answer 1

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⎟⎟ ✪✪ ＱＵＥＳＴＩＯＮ ✪✪ ⎟⎟

Give examples of polynomial p(x),g(x) and r(x), which satisfy the division algorithm and

deg r(x) = 0

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⎟⎟ ✰✰ ＡＮＳＷＥＲ ✰✰ ⎟⎟

$\tt{Let\:p(x)\:=\:2x^4\:+\:8x^3\:+\:6x^2\:4x\:+\:12}$

$\tt{r\:(x)\:=\:2}$

$\tt{On\:dividing\:2x^4\:+\:8x^3\:+\:6x^2\:4x\:+\:12\:by\:2}$

$\tt{we\:get}$

{HERE REFER THE IMAGE}

$\tt{Here,\:g(x)\:=\: x^4\:+\: 4x^3\:+\: 3x^2\: + \:2x \:+ \:1}$

$\tt{and\:r(x)\:=\:10\:so\:degree\:of\:r(x)\:=\:0}$

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Answer 2

||✪✪ QUESTION ✪✪||

Give examples of polynomial p(x),g(x) and r(x), which satisfy the division algorithm and deg r(x) = 0 ?

|| ✰✰ ANSWER ✰✰ ||

According to EUCLID division lemma : - a = bq + r where 0 ≤ r < b

So, Let P(x), g(x) , q(x), and r(x) satisfy EUCLID division lemma .

→ P(x) = g(x) × Q(x) + r(x) (where, 0≤ r(x) < g(x) )

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❶ when deg p(x) = deg q(x)

We know the formula :-

→ Dividend = Divisor x quotient + Remainder

So,

→ p(x) = g(x) × q(x) + r(x)

So, here the degree of quotient will be equal to degree of dividend when the divisor is constant.

Lets Assume The divison of 9x² + 6x + 3 by 3.

→ p(x) = 9x² + 6x + 3

→ g(x) = 3

→ q(x) = 3x² + 2x + 1

→ r(x) = 0

Degree of p(x) and q(x) is the same as 2.

Checking for division algorithm Now :-

→ p(x) = g(x) × q(x) + r(x)

→ (9x² + 6x + 3) = 3(3x² + 2x + 1) + 0

→ 9x² + 6x + 3 = 9x² + 6x + 3 + 0

→ 9x² + 6x + 3 = 9x² + 6x + 3

Hence, we can say That, the division algorithm is satisfied...

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❷ when deg q(x) = deg r(x)

Lets Assume The Division of 5x³ - 2x² + 12x by x².

→ p(x) = 5x³ - 2x² + 12x

→ g(x) = x²

→ q(x) = (5x - 2)

→ r(x) = 12x

Here , Degree of q(x) is same as Degree of r(x).

Checking for division algorithm Now :-

→ p(x) = g(x) × q(x) + r(x)

→ 5x³ - 2x² + 12x = x² × (5x - 2) + 12x

→ 5x³ - 2x² + 12x = 5x³ - 2x² + 12x

Hence, we can say That, the division algorithm is satisfied...

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❸ when deg r(x) = 0 .

Degree of remainder will be 0 when remainder comes to a constant.

Let us assume the division of 143x⁴ - 643x³ - 1226x² - 440x + 5072 by (x + 1).

→ p(x) = 143x⁴ - 643x³ - 1226x² - 440x + 5072 .

→ g(x) = (x + 1)

→ q(x) = (143x³ - 786x² - 440x)

→ r(x) = 5072

Here, Degree of r(x) is 0.

Checking for division algorithm Now :-

→ p(x) = g(x) × q(x) + r(x)

→ 143x⁴ - 643x³ - 1226x² - 440x + 5072 = (x + 1)(143x³ - 786x² - 440x) + 5072

→ 143x⁴ - 643x³ - 1226x² - 440x + 5072 = (143x⁴ - 786x³ - 440x² + 143x³ - 786x² - 440x) + 5072

→ 143x⁴ - 643x³ - 1226x² - 440x + 5072 = 143x⁴ - 643x³ - 1226x² - 440x + 5072 .

Hence, we can say That, the division algorithm is satisfied...

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