Question

Find the remainder When x^3+3x^2+3x+1
is divided by.
Q 1) x + 1


​

Answer 1

Step-by-step explanation:

By using remainder theorem :

• p(x) = x³ + 3x² + 3x + 1
• g(x) = x + 1

★ Calculating zero of the g(x) :

⇒ g(x) = 0

⇒ x + 1 = 0

⇒ x = 0 – 1

⇒ x = –1

➝ By using remainder theorem, we know that when p(x) is divided by (x + 1) then the remainder is p( –1).

So,

⇒ p(-1) = (-1)³ + 3(-1)² + 3(-1) + 1

⇒ p(-1) = -1 + 3(1) - 3 + 1

⇒ p(-1) = -1 + 3 - 3 + 1

⇒ p(-1) = 0

$\therefore$ The remainder is 0.

By using long division method :

Refer to attachment.

$\therefore$ The remainder is 0.

★ Some extra information :

What is long division method?

Long method of division is :-

$\large\boxed{\begin{array}{cc} \: \: \: \: \: \: \: \: \: \: \: \: \: \sf{Quotient} \\ \sf { Divisor\overline{) \; \;Dividend \; \;} } \\ \: \: \: \: \: \: \: \: \: \: \: \dots \dots \\ \: \: \: \: \: \: \: \: \: \: \dots \dots \\\sf{ \: \: \: \: \: \: \: \: \: \: \: \: \: Remainder} \end{array} }$

Where,

• Dividend = Divisor × Quotient + Remainder

Answer 2

By dividing (x + 1) from (x³ + 3x² + 3x + 1), we get quotient (x² + 2x + 1) and a zero remainder. {Through the process of long division (see attachment 01).}

ALITER:- {Remainder theorem}

Assume p(x) = x³ + 3x² + 3x + 1

And g(x) = x + 1

Clearly, zero of g(x) = (- 1).

So, p(- 1) = (- 1)³ + 3(- 1)² + 3(- 1) + 1

⇒ p(- 1) = - 1 + 3 - 3 + 1 = 0 (answer).

So, (x + 1) is a factor of (x³ + 3x² + 3x + 1).

Now, factoring the p(x):-

x³ + 3x² + 3x + 1

= x³ + x² + 2x² + 2x + x + 1

= x²(x + 1) + 2x(x + 1) + 1(x + 1)

= (x + 1)(x² + 2x + 1)

[x² + 2x + 1 is the quotient.]

ALITER:- {Synthetic Division}

(see attachment 02).

More:-

The division algorithm of polynominals states that:-

p(x) = g(x) · q(x) + r(x) where,

0 ≤ deg r(x) < deg g(x).

Relationship b/w zeroes, let those be α and βand numerical constants in a given quadratic polynomial in form of p(x) = ax² + bx + c:-

• α + β = - b/a
• αβ = c/a
• α - β = √(b² - 4ac)/2

Similarly, if f(x) is a cubic polynomial is form of ax³ + bx² + cx + d, and the zeroes are α, β, and γ,

• α + β + γ = - b/a
• αβ + βγ + αγ = c/a
• αβγ = - d/a.