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
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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 ?
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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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Verified answer
Step-by-step explanation:
[tex]{\Large{\mathbf{\orange{Question \: 1 :}}}} \\ [/tex]
➭ 8x² + 5x + 1 is divided by x - 10
[tex]{\large{\mathrm{\underline{\pink{Solution:}}}}} \\ [/tex]
Here, f(x) = 8x² + 5x + 1
By Remainder Theorem,
[tex]{\bold{\rm{⟼ \: \: \: \: \: The \: remainder \: when \: f(x) \: is \: divided \: by \: x \: - \: 10 \: is \: f(10)}}} \\ \\ [/tex]
[tex]{\bold{\sf{∴ \: \: \: \: \: remainder \: = \: f(10)}}} \\ \\ [/tex]
[tex]{\bold{\sf{⟼ \: \: \: \: \: f(10) \: = \: 8 \: × \: 10² \: + \: 5 \: × \: 10 \: + \: 1}}} \\ \\ [/tex]
[tex]{\bold{⟼ \: \: \: \: \: \: }}{\large{\mathrm{\underline{\fbox{\red{f(10) \: = \: 851}}}}}} \\ \\ \\[/tex]
[tex]{\Large{\mathbf{\orange{Question \: 2 :}}}} \\ [/tex]
➭ x² + 7x - 11 is divided by 3x - 2
[tex]{\large{\mathrm{\underline{\pink{Solution:}}}}} \\ [/tex]
Here, f(x) = x² + 7x - 11
[tex]{\bold{\sf{3x \: - \: 2 \: = \: 0}}} \\ [/tex]
[tex]{\bold{\sf{x \: = \: \frac{2}{3}}}} \\ [/tex]
By Remainder Theorem,
[tex]{\bold{\rm{⟼ \: \: \: \: \: The \: remainder \: when \: f(x) \: is \: divided \: by \: 3x \: - \: 2 \: is \: f \: \Big(\frac{2}{3}\Big)}}} \\ \\ [/tex]
[tex]{\bold{\sf{∴ \: \: \: \: \: remainder \: = \: f \Big(\frac{2}{3}\Big)}}} \\ \\ [/tex]
[tex]{\bold{\rm{⟼ \: \: \: \: \: f {\Big(\frac{2}{3}\Big) \: = \: \Big(\frac{2}{3}\Big)}^{2} \: + \: 7 \: \Big(\frac{2}{3}\Big) \: - \: 11}}} \\ \\ [/tex]
[tex]{\bold{\rm{⟼ \: \: \: \: \: \frac{4}{9} \: + \: \frac{14}{3} \: - \: 11}}} \\ \\ [/tex]
[tex] {\bold{⟼ \: \: \: \: \: }}{\large{\underline{\boxed{\mathrm{\red{- \: \: \frac{53}{9}}}}}}} \\ \\ \\ [/tex]
[tex]{\Large{\mathbf{\orange{Question \: 3 :}}}} \\ [/tex]
➭ 4x - 3x² + 2x - 4 is divided by x + 2
[tex]{\large{\mathrm{\underline{\pink{Solution:}}}}} \\ [/tex]
Here, f(x) = 4x - 3x² + 2x - 4
[tex]{\bold{\sf{x \: + \: 2 \: = \: 0}}} \\[/tex]
[tex]{\bold{\sf{x \: = \: -2}}} \\[/tex]
By Remainder Theorem,
[tex]{\bold{\rm{⟼ \: \: \: \: \: The \: remainder \: when \: f(x) \: is \: divided \: by \: x \: + \: 2 \: is \: f(-2)}}} \\ \\[/tex]
[tex]{\bold{\sf{∴ \: \: \: \: \: remainder \: = \: f(-2)}}} \\ \\ [/tex]
[tex]{\bold{\rm{⟼ \: \: \: \: \: 4(-2)³ \: - \: 3(-2)² \: + \: 2(-2) \: - \: 4}}} \\ \\[/tex]
[tex]{\bold{\rm{⟼ \: \: \: \: \: -32 \: - \: 12 \: - \: 4 \: - \: 4}}} \\ \\[/tex]
[tex]{\bold{⟼ \: \: \: \: \: \: }}{\large{\mathrm{\underline{\fbox{\red{f( - 2) \: = \: -52}}}}}} \\ [/tex]