Question

Help me :-

⭐ Refer attachment .
• Do only (iv) subdivision

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Answer 1

Answer:

Drinking 8 glasses of water a day is good for your health, If each glass of water

is equivalent to 148 mL, how many liters of water you need to consume

everyday.

Answer 2

★ Concept :-

Here the concept of Factor Theorem has been used. We see that we are given some polynomials and we have to check that if a given term is their factor or not. We can use remainder theorem for solving it which is the easiest way. In remainder theorem, firstly we take out the value of x from the divisor. Then we apply that value in the dividend to check if they are divisible by divisor or not if after application of value of x in dividend comes out to be zero.

Let's do it !!

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★ Solution :-

Given,

» Divisor = g(x) = x + 1

» Dividend = p(x)

By Factor Theorem, we know that

→ g(x) = 0

→ x + 1 = 0

→ x = -1

And we also know that, a expression is polynomial is the factor of g(x) when after application of the value of x in p(x) should be equal to 0.

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i.) x³ + x² + 1

$\;\sf{\rightarrow\;\;\pink{p(x)\;=\;\bf{x^{3}\;+\;x^{2}\;+\;1}}}$

By using Factor Theorem and applying value of x, we get

$\;\sf{\rightarrow\;\;p(-1)\;=\;\bf{(-1)^{3}\;+\;(-1)^{2}\;+\;1}}$

$\;\sf{\rightarrow\;\;p(-1)\;=\;\bf{(-1)(-1)(-1)\;+\;(-1)(-1)\;+\;1}}$

$\;\sf{\rightarrow\;\;p(-1)\;=\;\bf{-1\;+\;1\;+\;1}}$

$\;\sf{\rightarrow\;\;p(-1)\;=\;\bf{-1\;+\;2}}$

$\;\sf{\rightarrow\;\;p(-1)\;=\;\bf{-1\;+\;2}}$

$\;\bf{\rightarrow\;\;\pink{p(-1)\;=\;\bf{1}}}$

Here, p(x) = 1 and 1 ≠ 0.

Clearly, p(x) ≠ 0

This means (x + 1) is not the factor of this polynomial.

$\;\underline{\boxed{\tt{\purple{Hence,\;\:(x\:+\:1)\;is\;\:not\;\:the\;\:factor\;\:of\;\:this\;\:polynomial}}}}$

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ii) x⁴ + x³ + x² + x + 1

$\;\sf{\rightarrow\;\;\blue{p(x)\;=\;\bf{x^{4}\;+\;x^{3}\;+\;x^{2}\;+\;x\;+\;1}}}$

By using Factor Theorem and applying the value of x, we get

$\;\sf{\rightarrow\;\;p(-1)\;=\;\bf{(-1)^{4}\;+\;(-1)^{3}\;+\;(-1)^{2}\;+\;(-1)\;+\;1}}$

$\;\sf{\rightarrow\;\;p(-1)\;=\;\bf{(-1)(-1)(-1)(-1)\;+\;(-1)(-1)(-1)\;+\;(-1)(-1)\;+\;(-1)\;+\;1}}$

$\;\sf{\rightarrow\;\;p(-1)\;=\;\bf{1\;-\;1\;+\;1\;-\;1\;+\;1}}$

$\;\sf{\rightarrow\;\;p(-1)\;=\;\bf{-\:2\;+\;3}}$

$\;\bf{\rightarrow\;\;\blue{p(-1)\;=\;\bf{1}}}$

Here, p(x) = 1 and 1 ≠ 0

Clearly, p(x) ≠ 0

This means (x + 1) is not the factor of this polynomial.

$\;\underline{\boxed{\tt{\purple{Hence,\;\:(x\:+\:1)\;is\;\:not\;\:the\;\:factor\;\:of\;\:this\;\:polynomial}}}}$

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iii) x⁴ + 3x³ + 3x² + x + 1

$\;\sf{\rightarrow\;\;\red{p(x)\;=\;\bf{x^{4}\;+\;3x^{3}\;+\;3x^{2}\;+\;x\;+\;1}}}$

By using Factor Theorem and applying the value of x, we get

$\;\sf{\rightarrow\;\;p(-1)\;=\;\bf{(-1)^{4}\;+\;3(-1)^{3}\;+\;3(-1)^{2}\;+\;(-1)\;+\;1}}$

$\;\sf{\rightarrow\;\;p(-1)\;=\;\bf{(-1)(-1)(-1)(-1)\;+\;3(-1)(-1)(-1)\;+\;3(-1)(-1)\;+\;(-1)\;+\;1}}$

$\;\sf{\rightarrow\;\;p(-1)\;=\;\bf{1\;-\;3\;+\;3\;-\;1\;+\;1}}$

$\;\sf{\rightarrow\;\;p(-1)\;=\;\bf{-\:1\;+\;2}}$

$\;\bf{\rightarrow\;\;\red{p(-1)\;=\;\bf{1}}}$

Here p(x) = 1 and 1 ≠ 0

Clearly, p(x) ≠ 0

This means (x + 1) is not the factor of this polynomial.

$\;\underline{\boxed{\tt{\purple{Hence,\;\:(x\:+\:1)\;is\;\:not\;\:the\;\:factor\;\:of\;\:this\;\:polynomial}}}}$

______________________________

iv) x³ - x² - (2 + (-√2))x + √2

$\;\sf{\rightarrow\;\;\green{p(x)\;=\;\bf{x^{3}\;-\;x^{2}\;-\;(2\;+\;(-\sqrt{2}))x\;+\;\sqrt{2}}}}$

By using Factor Theorem and applying the value of x, we get

$\;\sf{\rightarrow\;\;p(-1)\;=\;\bf{(-1)^{3}\;-\;(-1)^{2}\;-\;(2\;-\;\sqrt{2})(-1)\;+\;\sqrt{2}}}$

$\;\sf{\rightarrow\;\;p(-1)\;=\;\bf{(-1)(-1)(-1)\;-\;(-1)(-1)\;-\;(-2\;+\;\sqrt{2})\;+\;\sqrt{2}}}$

$\;\sf{\rightarrow\;\;p(-1)\;=\;\bf{-\:1\;-\;(1)\;+\;2\;-\;\sqrt{2}\;+\;\sqrt{2}}}$

$\;\sf{\rightarrow\;\;p(-1)\;=\;\bf{-\:1\;-\;1\;+\;2\;-\;\sqrt{2}\;+\;\sqrt{2}}}$

$\;\sf{\rightarrow\;\;p(-1)\;=\;\bf{-\:2\;+\;2\;-\;\sqrt{2}\;+\;\sqrt{2}}}$

$\;\bf{\rightarrow\;\;\green{p(-1)\;=\;\bf{0}}}$

Here, p(x) = 0

Clearly, p(x) = 0

This means (x + 1) is the factor of this polynomial.

$\;\underline{\boxed{\tt{\purple{Hence,\;\:(x\:+\:1)\;is\;\:the\;\:factor\;\:of\;\:this\;\:polynomial}}}}$

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★ More to know :-

• Factor Theorem :: This theorem states that the polynomial p(x) which dividend results to 0 if the value of x is applied to it from the Divisor g(x) .

• Remainder Theorem :: This states that if a polynomial p(x) is divided by another polynomial g(x) then, then the non zero value obtained after applying the value of x from g(x) to p(x) is remainder when p(x) is divided by g(x).