Question

Please answer as soon as possible.
• TIME IS VERY LESS
• I NEED EXPLAINATION​

Answer 1

$\pink\bigstar$Answer:

• Option (A) Remainder = (-123)

$\blue\bigstar$ Given:

• On dividing x³ - 12x² - 42 by a polynomial x² - 9x - 27, the quotient obtained = x - 3

$\red\bigstar$To find:

• The remainder when x³ - 12x² - 42 is divided by x² - 9x - 27

$\green\bigstar$ Solution:

We know that :

• Dividend = Quotient × Divisor + Remainder

Here,

• Dividend = x³ - 12x² - 42
• Divisor = x² - 9x - 27
• Quotient = x - 3

So by substituting the given values, we get :

Dividend = Quotient × Divisor + Remainder

$\implies$x³ - 12x² - 42 = (x - 3)(x² - 9x - 27) + Remainder

[ By substituting the given values ]

$\implies$ x³ - 12x² - 42 = x³ - 9x² - 27x - 3x² + 27x + 81 + Remainder

$\implies$x³ - 12x² - 42 = x³ - 12x² + 81 + Remainder

$\implies$ x³ - 12x² - 42 - (x³ - 12x² + 81 ) = Remainder

[ Transposition Method ]

$\implies$ Remainder = x³ - 12x² - 42 - x³ + 12x² - 81

$\implies$ $\boxed{\sf Remainder\: = ( \:- \:123\:)}$

$\boxed{\sf \therefore Remainder\: = ( \: - \:123\:)}$

$\pink\bigstar$ Concepts Used:

• Dividend = Quotient × Divisor + Remainder
• Substitution of values
• Transposition Method
• Negative × Negative = Positive

$\blue\bigstar$ Extra - Information:

$\hookrightarrow$ Polynomials are the algebraic expressions which consist of variables and coefficients.

$\hookrightarrow$ The word polynomial is derived from the Greek words ‘poly’ means ‘many’ and ‘nominal’ means ‘terms’, so altogether it said “many terms”.

$\hookrightarrow$ The polynomial function is denoted by p(x) where x represents the variable. For example,

P(x) = x³ - x² + 11x - 17

If the variable is denoted by a, then the function will be p(a).