Question

If a polynomial x ki power 4 + 2 x cube + 8 x square + 12 x + 18 is divided by another polynomial X square + 5 the remainder comes out to be X + 2 find the value of P and Q

Answer 1

Appropriate Question :-

If the polynomial x⁴ + 2x³ + 8x² + 12x + 18 is divided by another polynomial x² + 5, the remainder comes out to be (px + q) then find the values of p and q.

Answer

Given :-

• The polynomial x⁴ + 2x³ + 8x² + 12x + 18 is divided by another polynomial x² + 5.

• The remainder comes out to be (px + q)

To Find:-

• The value of p and q.

Solution:-

Here, we first divide the polynomial x⁴ + 2x³ + 8x² + 12x + 18 is divided by another polynomial x² + 5, using long division.

So, by using long division, we have

$\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{}}}&{\underline{\sf{\: \: \: \: \: \: \: {x}^{2} \: + 2x \: + \: 3 \: \: \: \: \:\:}}}\\ {{\sf{ {x}^{2} + 5}}}& {\sf{\: {x}^{4} + {2x}^{3} + {8x}^{2} + 12x + 18 \:}} \\{\sf{}}&\underline{\sf{\: \: \: { - x}^{4} \: \: \: \: \: \: \: \: - 5{x}^{2} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:}}\\{\sf{}}&{\sf{\: \: \: \: \: \: \: \: \: \: \: {2x}^{3} + {3x}^{2} + 12x + 18\:\:}}\\{\sf{}}&\underline{\sf{\:\: { - 2x}^{3} \: \: \: \: \: \: \: \: - 10x \:\:}}\\{\sf{}}&{\sf{\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: {3x}^{2} + 2x + 18\:\: \: \: \: \: \: \: \: \: }}\\{\sf{}}& \underline{\sf{\:\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: { - 3x}^{2} \: \: \: \: \: \: \: \: - 15 \:\:}}\\{\sf{}}&\underline{\sf{\:\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 2x \: + \: 3 \:\:}}\end{array}\end{gathered}\end{gathered}\end{gathered}$

Thus,

$\rm :\longmapsto\:Remainder = 2x + 3$

But, it is given that

$\rm :\longmapsto\:Remainder = px + q$

So, on comparing we get,

$\bf\implies \:p = 2 \: \: \: \: \: \: and \: \: \: \: \: \: q = 3$