Question

Without actual division prove that x-2 is a factor of the polynomial 3x 3 -13x 2 +8x+12. using division or otherwise factorise it completely

Answer 1

$\underline {\blue {Without \: Actual \: division\: proving: }}$

$Let \: p(x) = 3x^{3} - 13x^{2} + 8x + 12\:and \\g(x) = x -2$

$The \: zero \:of \:g(x) \:is \: 2$

$Then \:p(-2) = 3(2)^{3} - 13(2)^{2} + 8 \times 2 + 12 \\= 3 \times (8) - 13 \times 4 + 16 + 12 \\= 24 - 52 + 16 + 12 \\= 52 - 52 \\= 0$

$So, \pink { by \:the \:Factor \: theorem }, \: (x-2) \:is \:a \\factor \: of \:3x^{3} - 13x^{2} + 8x + 12$

$\underline {\blue {Using \: division\: method: }}$

x-2|3x³-13x²+8x+12|3x²-7x-6

***** 3x^{3}-6x^{2}

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************ - 7x²+ 8x

************ -7x²+ 14x

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********************** - 6x + 12

********************** -6x + 12

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************* Remainder ( 0)

$Now, \: \blue { ( By\: division \: algorithm : )}\\</p><p>3x^{3} - 13x^{2} + 8x + 12\\ = (x-2)(3x^{2}-7x-6)\\= (x-2)( 3x^{2} -9x + 2x - 6 ) \\= (x-2)[3x(x-3) + 2(x-3) ]\\= (x-2)(x-3)(3x+2)$

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Answer 2

Answer:

(x minus 2) ( x minus 3) ( 3 X + 2)