Question

What must be added to \[3{x^3} + {x^2} - 22x + 9\;\] so that the result is exactly divisible by \[3{x^2} + 7x - 6\]?

Answer 1

Hint: We should know how to do division of polynomials using the long division method. And we know the dividend formula that is \[Dividend = divisor \times quotient + remainder\] .Remainder is subtracted from dividend to make it completely divisible by divisor.

Complete answer:
We have dividend= \[3{x^3} + {x^2} - 22x + 9\;\]
And divisor= \[3{x^2} + 7x - 6\]
By using long division method, we get,
\[3{x}^2+7x-6\overset{x-2}{\overline{\left){\begin{align} & {3{x}^{3}}+{{x}^{2}}-22x+9 \\ & \underline{{3{x}^{3}}+7{{x}^{2}-6x}} \\ & -6{{x}^{2}}-16x+9 \\ & \underline{-6{{x}^{2}}-14x+12} \\ & -2x-3 \\ \end{align}}\right.}}\]
When \[3{x^3} + {x^2} - 22x + 9\;\] is divided by \[3{x^2} + 7x - 6\]
We will get quotient as $x - 2$
and
Remainder \[ = \left( { - 2x - 3} \right)\]
We will \[ - (2x + 3)\] subtracted from \[3{x^3} + {x^2} - 22x + 9\;\] to get it completely divisible.
\[2x + 3\] must be added to \[3{x^3} + {x^2} - 22x + 9\;\] , to make the polynomial exactly divisible by \[3{x^2} + 7x - 6\]

So, the correct option is \[c)2x - 3\] .

Note:
We should know how to do division using the long division method. for doing division,
We should follow these steps: Double-check that the polynomial is written in ascending order. If any terms are missing, fill in the blanks with a zero (this will help with the spacing). Subtract the highest power word inside the division symbol from the highest power term outside the division symbol. Multiply (or distribute) the answer obtained in the previous step by the polynomial in front of the division symbol. Subtract from the previous term and bring it down. Continue using Steps 2, 3, and 4 until no more terms need to be brought down. Compose your final response. The residual is the phrase that remains after the final subtract step.