Question

polynomial 3x^3-5x^2+kx-2 and -x^3-x^2 + 7 x + k have the same remainder when divided by x + 2. find the value of k.

Answer 1

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Answer 2

Value of k is -12.

Given:

• $3 {x}^{3} - 5 {x}^{2} + kx - 2$ and $- {x}^{3} - {x}^{2} + 7x + k$
• Both have the same remainder when divided by (x+2).

To find:

• Find the value of k.

Solution:

Concept/Theorem to be used:

• Apply remainder theorem
• Remainder Theorem: It states that if p(x) is divided by (x-a), then p(a) will be the remainder.

Step 1:

Find value of x from divisor polynomial.

$(x + 2) = 0$

or

$\bf x = - 2$

Step 2:

Put x= -2 in first polynomial; let the first polynomial is p1(x).

$p_1( - 2) = 3 {( - 2)}^{3} - 5 {( - 2)}^{2} + k( - 2) - 2$

or

$p_1( - 2) = - 24 - 20 - 2k - 2$

or

$\bf p_1( - 2) = - 46 - 2k ...eq1$

Step 3:

Put x=-2 in second polynomial p2(x).

$p_2( - 2) = - {( - 2)}^{3} - {( - 2)}^{2} + 7( - 2) + k$

or

$p_2( - 2) = 8 - 4 - 14 + k$

or

$\bf p_2( - 2) = k - 10...eq2$

Step 4:

Equate both equations.

$\bf p_1( - 2) = p_2( - 2)$

or

$- 46 - 2k = k - 10$

or

$- 2k - k = - 10 + 46$

or

$- 3k = 36$

or

$k = - \frac{36}{3}$

or

$\bf \red{k = - 12}$

Thus,

Value of k is -12.

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