show all algebraic work for full credit.
1) use the leading coefficient and degree of the polynomial function ( f(x)=x^{3}-7 x^{2}+10 x ) to determine the end behavior of the graph. make sure to use the correct notation.
2) the graph of a function, ( f ), is shown below. use the graph to estimate all turning points. circle all ( x )-values that are turning points of ( f ).
3) multiply the following polynomials and write the answer in standard form.
( (x + 2 y)left(x^{2}-x y+3 y
ight) )
4) multiple choice (one answer): select the graph of a polynomial function ( f(x) ) that satisfies the following clues:
( f(x) ) is positive on the intervals ( (-infty,-3),(-2,0) ), and ( (2,3) ).
( f(x) ) is negative on the intervals ( (-3,-2),(0,2) ), and ( (3, infty) ).
( f(x) ) is increasing on the intervals ( (-2.59,-1.03) ) and ( (1.03,2.59) ).
( f(x) ) is decreasing on the intervals ( (-infty,-2.59),(-1.03,1.03) ), and ( (2.59, infty) ).

Question

Accepted Answer

### 1)
# Explanation:
## Step1: Identify the leading coefficient and degree
The polynomial function is $f(x)=x^{3}-7x^{2}+10x$. The leading term is $x^{3}$, so the leading coefficient $a = 1$ (positive) and the degree $n=3$ (odd).
## Step2: Determine the end - behavior
For a polynomial function $y = a x^{n}$ with $a>0$ and $n$ odd:
As $x
ightarrow-\infty$, $y = f(x)
ightarrow-\infty$ (since $y=1\times(-\infty)^{3}=-\infty$)
As $x
ightarrow+\infty$, $y = f(x)
ightarrow+\infty$ (since $y = 1\times(+\infty)^{3}=+\infty$)

# Answer:
As $x
ightarrow-\infty$, $f(x)
ightarrow-\infty$; as $x
ightarrow+\infty$, $f(x)
ightarrow+\infty$

### 2)
# Explanation:
A turning point of a function is a point where the function changes from increasing to decreasing or vice - versa.
Looking at the graph:
- At $x=-1.5$, the function changes from decreasing to increasing.
- At $x = 1.5$, the function changes from increasing to decreasing.

# Answer:
d) $-1.5$, f) $1.5$

### 3)
# Explanation:
## Step1: Use the distributive property (FOIL - like for polynomials with more than two terms)
$(x + 2y)(x^{2}-xy + 3y)=x(x^{2}-xy + 3y)+2y(x^{2}-xy + 3y)$
## Step2: Expand each part
$x(x^{2}-xy + 3y)=x^{3}-x^{2}y+3xy$
$2y(x^{2}-xy + 3y)=2x^{2}y-2xy^{2}+6y^{2}$
## Step3: Combine like terms
$x^{3}-x^{2}y + 3xy+2x^{2}y-2xy^{2}+6y^{2}=x^{3}+( - x^{2}y+2x^{2}y)-2xy^{2}+3xy + 6y^{2}=x^{3}+x^{2}y-2xy^{2}+3xy + 6y^{2}$

# Answer:
$x^{3}+x^{2}y-2xy^{2}+3xy + 6y^{2}$

### 4)
Since no graph options (A, B, C, D) are provided in a machine - readable format, we can't give a definite answer. But the general approach would be:
# Explanation:
- Check the sign of the function ($y = f(x)$) in the given intervals. For example, in $(-\infty,-3)$ the function is positive (above the $x$ - axis), in $(-3,-2)$ it is negative (below the $x$ - axis).
- Check the increasing/decreasing nature. In $(-2.59,-1.03)$ the function is increasing (slope positive), in $(-1.03,1.03)$ it is decreasing (slope negative)

# Answer:
Without the actual graph images (A, B, C, D) in a proper format, we can't select the exact graph. But the graph should satisfy: positive in $(-\infty,-3)\cup(-2,0)\cup(2,3)$, negative in $(-3,-2)\cup(0,2)\cup(3,\infty)$, increasing in $(-2.59,-1.03)\cup(1.03,2.59)$ and decreasing in $(-\infty,-2.59)\cup(-1.03,1.03)\cup(2.59,\infty)$

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QUESTION IMAGE

Question

Explanation:

1)

Step1: Identify the leading coefficient and degree

Step2: Determine the end - behavior

Step1: Use the distributive property (FOIL - like for polynomials with more than two terms)

Step2: Expand each part

Step3: Combine like terms

Answer:

2)