Question

What must be added to the polynomial p(x) =$2{{x}^{3}}+9{{x}^{2}}-5x-15$, so that the resulting polynomial is exactly divisible by 2x + 3.

Answer 1

Hint: First, realise that we must add a constant number to the polynomial p(x) so that it is exactly divisible by 2x + 3. Let c be this constant number. Then use the remainder theorem: The remainder theorem states that when a polynomial, f(x), is divided by a linear polynomial, x - a, the remainder of that division will be equivalent to f(a). using this, we get $P\left( \dfrac{-3}{2} \right)+c=0$. Now, solve for c which is our final answer.

Complete step-by-step answer:
In this question, we need to find what must be added to the polynomial p(x) =$2{{x}^{3}}+9{{x}^{2}}-5x-15$, so that the resulting polynomial is exactly divisible by 2x + 3.
The step by step explanation is given below:

We are given the polynomial p(x) =$2{{x}^{3}}+9{{x}^{2}}-5x-15$

We have to realise that we must add a constant number to the polynomial p(x) so that it is exactly divisible by 2x + 3.

Let c be the constant number which must be added to the above polynomial so that it is exactly divisible by 2x + 3.

We will use the remainder theorem.

The remainder theorem states that when a polynomial, f(x), is divided by a linear polynomial, x - a, the remainder of that division will be equivalent to f(a). In other words, if you want to evaluate the function f(x) for a given number, a, you can divide that function by x - a and your remainder will be equal to f(a).

Here, we have a = $\dfrac{-3}{2}$

Using the remainder theorem, we will have the following:

$P\left( \dfrac{-3}{2} \right)+c=0$

$2{{\left( \dfrac{-3}{2} \right)}^{3}}+9{{\left( \dfrac{-3}{2} \right)}^{2}}-5\left( \dfrac{-3}{2}

\right)-15+c=0$

$c+6=0$

$c=-6$

Hence, -6 must be added to the polynomial p(x) =$2{{x}^{3}}+9{{x}^{2}}-5x-15$, so that the resulting polynomial is exactly divisible by 2x + 3.

This is our final answer.

Note: In this question, it is very important to realise that we must add a constant number to the polynomial p(x) so that it is exactly divisible by 2x + 3 and that the remainder theorem states that when a polynomial, f(x), is divided by a linear polynomial, x - a, the remainder of that division will be equivalent to f(a).