p(x)=x3+8x2−7x+12 and g(x)=x−1. If p(x) is divided by g(x), it giv

Question

p (x) = x^{3} + 8 x^{2} - 7 x + 12

and

g (x) = x - 1

. If

p (x)

is divided by

g (x)

, it gives

q (x)

and

r (x)

as quotient and remainder respectively. If

a

is the degree of

q (x)

and

b

is the degree of

r (x)

(a - b) = ?

Options:

A . 1

B . 2

C . 3

D . 4

Answer: Option B
:
B
Remainder

= r (x) = p (1) = 1^{3} + 8 \times 1^{2} - 7 \times 1 + 12 = 14

The degree of a polynomial is the highest degreeof its terms when the polynomial is expressed in its canonical form consisting of a linear combination of monomials.
So, Degree of

r (x) = b = 0

q (x) = \frac{p (x)}{g (x)}

Now ,

p (x) = x^{3} + 8 x^{2} - 7 x + 12

g (x) = (x - 1)

Performing long division

x^{2} + 9 x + 2 x - 1) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ x^{3} + 8 x^{2} - 7 x + 12 x^{3} - x^{2} (-) (+) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 9 x^{2} - 7 x 9 x^{2} - 9 x (-) (+) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ + 2 x + 12 2 x - 2 (-) (+) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 14

∴ q (x) = \frac{p (x)}{g (x)} = x^{2} + 9 x + 2

So, degree of

q (x) = a = 2

Hence,

a - b = 2 - 0 = 2

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