Question

Apply the division algorithm to find the quotient and remainder on dividing f(x) = x3-6x2+11x-6 by g(x) = x+2​

Answer 1

Answer:

Quotient=x²-8x+27

Remainder=-60

For EXPLANATION, see the attachment

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Answer 2

Answer:

GIVEN :

• $\sf f(x) \: = \: x^{3} \: - \: 6x^{2} \: + \: 11x \: - \: 6$

• $\sf g(x) \: = \: x \: + \: 2$

TO FIND :

Quotient = ?

Remainder = ?

SOLUTION :

Using division algorithm,

$\sf f(x) \: = \: x^{3} \: - \: 6x^{2} \: + \: 11x \: - \: 6$

→ $x^{3} \: - \: 6x^{2} \: 11x \: - \: 6 \: =</p><p> \: (x \: + \: 2) \: (ax^{2} \: + \: bx \: + \: c)</p><p> \: + \: k x^{3} \: - \: 6x^{2} </p><p>\: + \: 11x \: - \: 6 \: = \: [x(ax^{2} \: + \: bx \: + c) \: + \: 2 (ax^{2} \: + \: bx \: + c)]+k$

→ $x^{3} \: - \: 6x^{2} \: + \: 11x \: - \: 6 \: = (b \: + \: 2a)x^{2} \: + \: (c \: + \: 2b)x \: + \: (2c \: + \: k)$

Now, comparing the coefficients of the variables,

• $\sf {a \: = \: 1}$

→ $\sf {b \: + \: 2a \: = \: -6}$

→ $\sf {b \: + \: (2 \times 1) \: = \: -6}$

→ $\sf {b \: = \: -6 \: - \: 2}$

→ $\sf \boxed {b \: = \: -8}$

→ $\sf {c \: + \: 2b \: = \: 11}$

→ $\sf {c \: + \: (2 \times -8) \: = \: 11}$

→ $\sf {c \: - 16 \: = \: 11}$

→ $\sf {c \: = \: 11 \: + \: 16}$

→ $\sf \boxed {c \: = \: 27}$

→ $\sf {2c \: + \: k \: = \: -6}$

→ $\sf {2 \: \times \: 27 \: + \: k \: = \: -6}$

→ $\sf {54 \: + \: k \: = \: -26}$

→ $\sf \boxed {k \: = \: -60}$

Substituting the values in equation to get quotient and remainder,

→ $\sf {q(x) \: ax^{2} \: + \: bx \: + c}$

→ $\sf \boxed {q(x) \: = \: x^{2} \: - \: 8x \: + \: 27}$

→ $\sf {r(x) \: = \: k}$

→ $\sf \boxed {r(x) \: = \: -60}$