Question

Use synthetic division to determine whether the number $k$ is an upper or lower bound (as specified) for the real zeros of the function $f.$

$k=4;f\left(x\right)=2x^3-2x^2-3x-5;$ Lower bound?

Answer 1

Find whether $\mathit{K}\mathbf{=}\mathbf{4}$ is lower bound or not:

Given that $\begin{array}{rcl}f\left(x\right)& =& 2x^3-2x^2-3x-5\end{array}$ and the bound value $k=4$

Step- 1: Apply the synthetic division method:

In general, the synthetic division is used to find the zeroes of polynomials when dividing a polynomial by a polynomial equation of degree $1.$

In the function, $\begin{array}{rcl}f\left(x\right)& =& 2x^3-2x^2-3x-5\end{array}$ the co-efficient of $x$ is the dividend $\Rightarrow 2-2-2-5$

The bound value of $k$ is the divisor which has the degree $1$ $\Rightarrow 4^1$

Place the numbers representing the divisor and the dividend into a division-like configuration:

$\begin{array}{lllll}4& 2& -2& -3& -5\\ & & & & \end{array}$

Step- 2: The first result should always be the first dividend $\mathbf{\left(}\mathbf{2}\mathbf{\right)}$:

$\begin{array}{lllll}4& 2& -2& -3& -5\\ & 2& & & \end{array}$

Step- 3: Multiply the result $\mathbf{\left(}\mathbf{2}\mathbf{\right)}$ by the divisor $\mathbf{\left(}\mathbf{4}\mathbf{\right)}$:

$\begin{array}{lllll}4& 2& -2& -3& -5\\ & & 8& & \\ & 2& & & \end{array}$

Step- 4: Now, add the result $\mathbf{\left(}\mathbf{8}\mathbf{\right)}$ with the next coefficient of the dividend $\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{)}$:

$\begin{array}{lllll}4& 2& -2& -3& -5\\ & & 8& & \\ & 2& 6& & \end{array}$

Step- 5: Multiply the root $\mathbf{\left(}\mathbf{4}\mathbf{\right)}$ by the last result $\mathbf{\left(}\mathbf{6}\mathbf{\right)}$:

$\begin{array}{lllll}4& 2& -2& -3& -5\\ & & 8& 24& \\ & 2& 6& & \end{array}$

Step- 6: Add the result $\mathbf{\left(}\mathbf{24}\mathbf{\right)}$ with the next coefficient of the dividend $\mathbf{(}\mathbf{-}\mathbf{3}\mathbf{)}$:

$\begin{array}{lllll}4& 2& -2& -3& -5\\ & & 8& 24& \\ & 2& 6& 21& \end{array}$

Step- 7: Multiply the root $\mathbf{\left(}\mathbf{4}\mathbf{\right)}$ by the last result $\mathbf{\left(}\mathbf{21}\mathbf{\right)}$:

$\begin{array}{lllll}4& 2& -2& -3& -5\\ & & 8& 24& 81\\ & 2& 6& 21& \end{array}$

Step- 8: Add the result $\mathbf{84}$ to the next coefficient of the dividend $\mathbf{(}\mathbf{-}\mathbf{5}\mathbf{)}$:

$\begin{array}{lllll}4& 2& -2& -3& -5\\ & & 8& 24& 81\\ & 2& 6& 21& 79\end{array}$

Thus the obtained coefficients or the zero polynomials are $2,6,21,79.$ All the zero polynomials are positive. So, $k=4$ is the upper root.

Hence, for the function $\begin{array}{rcl}\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{}& \mathbf{=}& \mathbf{}\mathbf{2}{\mathit{x}}^{\mathbf{3}}\mathbf{-}\mathbf{2}{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{3}\mathit{x}\mathbf{-}\mathbf{5}\end{array}$ ,$\mathit{k}\mathbf{=}\mathbf{4}$ is the upper bound.