Question

What must be subtracted from the polynomial \[{x^3} + 13{x^2} + 35x + 25\] so that the resulting polynomials is exactly divisible by \[{x^2} + 11x + 10\].

Answer 1

Hint: Here, we have to find the polynomial which has to be subtracted from the given polynomial. A polynomial is an expression consisting of both variables and coefficients, that involves only the mathematical operations and non-negative integer exponentiation of variables. Here we will divide the polynomials and find the remainder. We will then subtract the remainder from the dividend to get the resulting polynomial.

Complete step-by-step answer:
Let \[P(x) = {x^3} + 13{x^2} + 35x + 25\] and \[Q(x)\] be the resulting polynomial subtracted from \[P(x)\] .
Now, we will divide the polynomial \[P(x)\] by \[{x^2} + 11x + 10\].
So, here \[P(x)\] is the dividend and \[{x^2} + 11x + 10\] is the divisor.
Therefore, dividing the polynomials, we get
\[{x^2} + 11x + 10\mathop{\left){\vphantom{1\begin{array}{l}{x^3} + 13{x^2} + 35x + 25\\\underline {{x^3} + 11{x^2} + 10x} \\00 + 2{x^2} + 25x + 25\\\underline {00 + 2{x^2} + 22x + 20} \\\underline {00 + 00 + 3x + 5} \end{array}}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{\begin{array}{l}{x^3} + 13{x^2} + 35x + 25\\\underline {{x^3} + 11{x^2} + 10x} \\00 + 2{x^2} + 25x + 25\\\underline {00 + 2{x^2} + 22x + 20} \\\underline {00 + 00 + 3x + 5} \end{array}}}}
\limits^{\displaystyle\,\,\, {x + 2}}\]
Here, we get the quotient as \[x + 2\] and remainder as \[3x + 5\].
So, here the remainder has to be subtracted from the polynomial \[P(x)\].
\[Q(x) = P(x) - \] Remainder
Substituting the values of \[P(x) = {x^3} + 13{x^2} + 35x + 25\] and the remainder, we have
\[ \Rightarrow Q(x) = {x^3} + 13{x^2} + 35x + 25 - \left( {3x + 5} \right)\]
Rewriting the equation, we have
\[ \Rightarrow Q(x) = {x^3} + 13{x^2} + 35x + 25 - 3x - 5\]
Adding and subtracting the like terms, we have
\[ \Rightarrow Q(x) = {x^3} + 13{x^2} + 32x + 20\]
So, the resulting polynomial is \[Q(x) = {x^3} + 13{x^2} + 32x + 20\]
Therefore, \[3x + 5\] is the polynomial which has to be subtracted from the polynomial \[{x^3} + 13{x^2} + 35x + 25\] so that it is exactly divisible by \[{x^2} + 11x + 10\]. Thus the resulting polynomial \[Q(x) = {x^3} + 13{x^2} + 32x + 20\] would be exactly divisible by \[{x^2} + 11x + 10\].

Note: We can verify the answer by dividing the resulting polynomial by the divisor. If the remainder is zero, then the resulting polynomial would be exactly divisible by the divisor. If the remainder is not equal to zero, then the resulting polynomial is not divisible by the divisor. If we want to find the polynomial which must be added to the given polynomial to get exactly divisible by the divisor, then the remainder has to be added to the given polynomial.