Question

The remainder when 4x^3- 3x^2+2=0 divided by (x-2) is...​

Answer 1

Correct Question:

Find the remainder when the polynomial 4x³ - 3x² + 2 is divided by (x - 2) is...

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Answer:

• Remainder is 22.

Solution:

Method (1) :

Long division method.

x - 2 ) 4x³ - 3x² + 2 ) 4x² + 5x + 10x

⠀⠀⠀⠀4x³ - 8x²

⠀⠀⠀___________

⠀⠀⠀⠀⠀⠀⠀5x² + 2

⠀⠀⠀⠀⠀⠀⠀5x² - 10x

⠀⠀⠀________________

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀10x + 2

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀10x - 20

⠀⠀⠀___________________

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀22

Remainder = 22

Verification:

Dividend = Quotient × Divisor + Remainder

R.H.S. = (4x² + 5x + 10) (x - 2) + 22

⠀⠀⠀⠀= 4x³ + 5x² + 10x - 8x² - 10x - 20 + 22

⠀⠀⠀⠀= 4x³ - 3x² + 2

⠀⠀⠀⠀= L.H.S.

Hence, verified.

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Method (2) :

Remainder theorem.

If p (x) = 4x³ - 3x² + 2 divided by f (x) = x - 2, and f (x) is in the form of $\sf{x-\alpha}$ then remainder can be given by p $\sf{(\alpha)}$.

Here, comparing x - 2 with x - $\sf{\alpha}$

we get $\sf{\alpha}$ = 2

Remainder = f (2)

⠀⠀⠀⠀⠀⠀⠀= 4(2)³ - 3(2)² + 2

⠀⠀⠀⠀⠀⠀⠀= 4(8) - 3(4) + 2

⠀⠀⠀⠀⠀⠀⠀= 32 - 12 + 2

⠀⠀⠀⠀⠀⠀⠀= 22

Therefore, remainder is 22.