Question

find the remainder when 2 x cube +5 x square -6x+2 is divided by x+2​

Answer 1

Hey,

　

Question :-

Find the remainder when 2x³ + 5x² - 6x + 2 is divided by x+ 2

　

Solution :-

f(x) = x + 2 = 0

f(x) = x = 0 - 2

f(x) = x = -2

f(x) = -2

p(x) = 2x³ + 5x² - 6x + 2

Putting the value -2 in place of x

p(-2) = 2(-2)³ + 5(-2)² - 6(-2) + 2

p(-2) = -16 + 20 + 12 + 2

p(-2) = 18ㅤ

　

Further Information :-

1. Let f(x) be a polynomial of degree x > 1 and let a be any real number. When f(x) is divided by (x - a), then the remainder is f(a). (Remainder Theorem)

2. Let f(x) be a polynomial of degree greater than or equal to 1 and a be any real number such that p(a) = 0, then (x - a) is a factor of f(x). (Factor Theorem)

3. An algebraic expression in which the variables involved have oy non - negative integral powers is called a polynomial. (Non - negative integrals :- 0, 1, 2, 3, 4, 5...) whose powers are not in negative or fraction.

　

Hope That Helps :)

@MagicHeart~

Answer 2

$\texttt{\textsf{\large{\underline{Solution}:}}}$

Let:

$\sf \implies p(x) = 2 {x}^{3} + 5 {x}^{2} - 6x + 2$

We have to find out the remainder when p(x) is divided by x + 2.

Therefore, as per remainder theorem:

$\sf \implies Remainder = p( - 2)$

$\sf \implies p( - 2) = 2 \times {( - 2)}^{3} + 5 \times {( - 2)}^{2} - 6 \times ( - 2) + 2$

$\sf \implies p( - 2) = - 16 +20 + 12 + 2$

$\sf \implies p( - 2) = 4+ 12 + 2$

$\sf \implies p( - 2) = 6 + 12$

$\sf \implies p( - 2) = 18$

★ So, the remainder when p(x) is divided by (x + 2) is 18.

$\texttt{\textsf{\large{\underline{Answer}:}}}$

• The remainder when the given polynomial is divided by (x + 2) is 18.

$\texttt{\textsf{\large{\underline{Concept}:}}}$

• Remainder Theorem: If a polynomial f(x) is divided by (x - α), then remainder = f(α).
• Similarly, if a polynomial f(x) is divided by (x + α) i.e., (x - (-α)), then remainder = f(-α).
• Using remainder theorem, problem is solved.