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Q. Is 3x+2 a factor of 6x(raised to the power)3+7x(raised to the power) 2-x-2. Show that DIVIDEND=DIVISOR*QUOTIENT+REMAINDER
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3 x + 2 is factor of P(x) = 6 x³ + 7 x² - x - 2 ??
If 3 (x + 2/3) is a factor of the expression, P(x = - 2/3) should be 0.
Reminder of division of P(x) by (x+2/3) is P( - 2/3)
P(2/3) = -48/27 + 28/9 + 2/3 - 2 = 0
As the reminder is zero, (x + 2/3) is a factor.
Hence, 6 x³ + 7 x² - x - 2 = 3 (x + 2/3) ( a x² + b x + c) + 0
comparing coefficients of x³ on both sides, 3 a = 6 => a = 2
coefficients of the constant terms: -2 = 2 c => c = -1
finding coefficient of x² term: 7 = 2 a + 3 b => b = 1
quotient : 2 x² + x - 1
==========================================
3 x + 2 ) 6 x³ + 7 x² - x - 2 ( 2 x² + x - 1
6 x³ + 4 x²
============
3 x² - x
3 x² + 2 x
===========
- 3 x - 2
- 3 x - 2
=======
0
If 3 (x + 2/3) is a factor of the expression, P(x = - 2/3) should be 0.
Reminder of division of P(x) by (x+2/3) is P( - 2/3)
P(2/3) = -48/27 + 28/9 + 2/3 - 2 = 0
As the reminder is zero, (x + 2/3) is a factor.
Hence, 6 x³ + 7 x² - x - 2 = 3 (x + 2/3) ( a x² + b x + c) + 0
comparing coefficients of x³ on both sides, 3 a = 6 => a = 2
coefficients of the constant terms: -2 = 2 c => c = -1
finding coefficient of x² term: 7 = 2 a + 3 b => b = 1
quotient : 2 x² + x - 1
==========================================
3 x + 2 ) 6 x³ + 7 x² - x - 2 ( 2 x² + x - 1
6 x³ + 4 x²
============
3 x² - x
3 x² + 2 x
===========
- 3 x - 2
- 3 x - 2
=======
0
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