Question

Find the remainder (without division) when :
(a) 8x² + 5x + 1 is divided by x - 10.
(b) x² + 7x - 11 is divided by 3x-2
(c) 4x - 3x² + 2x - 4 is divided by x + 2

Is any Moderator here to solve this ? ​

Answer 1

Answer:

(a) 10

(b) -539

(c) -52

Step-by-step explanation:

(a) Here, f(x) = 8x2 + 5x + 1.

By remainder Theorem,

The remainder when f(x) is divided by x – 10 is f(10).

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(b)  Here, f(x) = x2 + 7x – 11 and 3x - 2 = 0 ⟹  x = 23

By remainder Theorem,

The remainder when f(x) is divided by 3x - 2 is f(23).

Therefore, remainder = f(23) = (23)2 + 7 ∙ (23) - 11

= 49 + 143 - 11

= -539

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(c) Here, f(x) = 4x3 - 3x2 + 2x - 4 and x + 2 = 0 ⟹  x = -2

By remainder Theorem,

The remainder when f(x) is divided by x + 2 is f(-2).

Therefore, remainder = f(-2) = 4(-2)3 - 3 ∙ (-2)2 + 2 ∙ (-2) - 4

= - 32 - 12 - 4 - 4

= -52

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Answer 2

Step-by-step explanation:

${\Large{\mathbf{\orange{Question \: 1 :}}}}$

➭ 8x² + 5x + 1 is divided by x - 10

${\large{\mathrm{\underline{\pink{Solution:}}}}}$

Here, f(x) = 8x² + 5x + 1

By Remainder Theorem,

${\bold{\rm{⟼ \: \: \: \: \: The \: remainder \: when \: f(x) \: is \: divided \: by \: x \: - \: 10 \: is \: f(10)}}}$

${\bold{\sf{∴ \: \: \: \: \: remainder \: = \: f(10)}}}$

${\bold{\sf{⟼ \: \: \: \: \: f(10) \: = \: 8 \: × \: 10² \: + \: 5 \: × \: 10 \: + \: 1}}}$

${\bold{⟼ \: \: \: \: \: \: }}{\large{\mathrm{\underline{\fbox{\red{f(10) \: = \: 851}}}}}}$

${\Large{\mathbf{\orange{Question \: 2 :}}}}$

➭ x² + 7x - 11 is divided by 3x - 2

${\large{\mathrm{\underline{\pink{Solution:}}}}}$

Here, f(x) = x² + 7x - 11

${\bold{\sf{3x \: - \: 2 \: = \: 0}}}$

${\bold{\sf{x \: = \: \frac{2}{3}}}}$

By Remainder Theorem,

${\bold{\rm{⟼ \: \: \: \: \: The \: remainder \: when \: f(x) \: is \: divided \: by \: 3x \: - \: 2 \: is \: f \: \Big(\frac{2}{3}\Big)}}}$

${\bold{\sf{∴ \: \: \: \: \: remainder \: = \: f \Big(\frac{2}{3}\Big)}}}$

${\bold{\rm{⟼ \: \: \: \: \: f {\Big(\frac{2}{3}\Big) \: = \: \Big(\frac{2}{3}\Big)}^{2} \: + \: 7 \: \Big(\frac{2}{3}\Big) \: - \: 11}}}$

${\bold{\rm{⟼ \: \: \: \: \: \frac{4}{9} \: + \: \frac{14}{3} \: - \: 11}}}$

${\bold{⟼ \: \: \: \: \: }}{\large{\underline{\boxed{\mathrm{\red{- \: \: \frac{53}{9}}}}}}}$

${\Large{\mathbf{\orange{Question \: 3 :}}}}$

➭ 4x - 3x² + 2x - 4 is divided by x + 2

${\large{\mathrm{\underline{\pink{Solution:}}}}}$

Here, f(x) = 4x - 3x² + 2x - 4

${\bold{\sf{x \: + \: 2 \: = \: 0}}}$

${\bold{\sf{x \: = \: -2}}}$

By Remainder Theorem,

${\bold{\rm{⟼ \: \: \: \: \: The \: remainder \: when \: f(x) \: is \: divided \: by \: x \: + \: 2 \: is \: f(-2)}}}$

${\bold{\sf{∴ \: \: \: \: \: remainder \: = \: f(-2)}}}$

${\bold{\rm{⟼ \: \: \: \: \: 4(-2)³ \: - \: 3(-2)² \: + \: 2(-2) \: - \: 4}}}$

${\bold{\rm{⟼ \: \: \: \: \: -32 \: - \: 12 \: - \: 4 \: - \: 4}}}$

${\bold{⟼ \: \: \: \: \: \: }}{\large{\mathrm{\underline{\fbox{\red{f( - 2) \: = \: -52}}}}}}$