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We are given a polynomial function $f\left(x\right)=x^4+3x^3-5x^2+9$.

We need to find the bounds to the zeros of the function and using it we need to obtain its graph.

Letf denote a polynomial function whose leading coefficient is $1$.

$f\left(x\right)=x^n+a_{n-1}{x}^{n-1}+...+a_{1}x+a_0$

A bound $M$on the real zeros of f is the smaller of the two numbers

$Max\left\{1,\left|a_0\right|+\left|a_1\right|+...+\left|a_{n-1}\right|\right\},1+Max\left\{\left|a_0\right|,\left|a_1\right|,...,\left|a_{n-1}\right|\right\}$

where Max $\left\{\right\}$means 鈥渃hoose the largest entry in $\left\{\right\}$.鈥�

For the given function $f\left(x\right)=x^4+3x^3-5x^2+9$the leading coefficient is one, so on comparing it with standard form we get

$$\begin{gathered} a_3=3 \\ {a}_2=-5 \\ {a}_1=0 \\ {a}_0=9 \end{gathered}$$

Using the formula the two numbers are found as

$\begin{array}{rcl}Max\left\{1,\left|a_0\right|+\left|a_1\right|+\left|a_2\right|+\left|a_3\right|\right\}& =& Max\left\{1,\left|9\right|+\left|0\right|+\left|-5\right|+\left|3\right|\right\}\\ & =& Max\left\{1,9+5+3\right\}\\ & =& Max\left\{1,17\right\}\\ & =& 17\end{array}$

and

$\begin{array}{rcl}1+Max\left\{\left|a_0\right|,\left|a_1\right|,\left|a_2\right|,\left|a_3\right|\right\}& =& 1+Max\left\{\left|9\right|,\left|0\right|,\left|-5\right|,\left|3\right|\right\}\\ & =& 1+Max\left\{9,0,5,3\right\}\\ & =& 1+9\\ & =& 10\end{array}$

Among the two numbers $17$and $10$, $10$is the smallest.

Therefore, role="math" localid="1646137266818" $10$is the bound.

Every real zero off lies between $-10$and $10$.

Using the bound the graph of the function is given as