Math / Algebra

QuestionNAME DATE PERIOD
7. For each polynomial (a) write the degree (b) leading coefficient (c) zeros (d) describe the end behavior and (e) sketch a possible graph. You can then check your sketch using graphing technology. \begin{tabular}{|l|c|c|} \hline \multicolumn{3}{|c|}{ $A(x)=-(x+4)(x+1)(x-3)(x-8)$ } \\ \hline Degree & Leading Coefficient & Zeros \\ \hline \end{tabular}
End Behavior As x gets larger and larger negative, $\mathrm{A}(\mathrm{x})$ gets larger and larger $\qquad$ .
As $x$ gets larger and larger positive, $\mathrm{A}(\mathrm{x})$ gets larger and larger $\qquad$

Studdy Solution

STEP 1

What is this asking? We need to find some key features of a polynomial $A(x)$ like its degree, leading coefficient, zeros, and end behavior, and then sketch its graph! Watch out! Don't mix up the leading coefficient and the constant term.
Also, remember the connection between the zeros and the factors of the polynomial.

STEP 2

1. Find the Degree
2. Find the Leading Coefficient
3. Find the Zeros
4. Describe the End Behavior
5. Sketch the Graph

STEP 3

The degree of a polynomial is the highest power of $x$ when it's fully expanded.
Our polynomial $A(x) = -(x+4)(x+1)(x-3)(x-8)$ is in factored form.

STEP 4

Imagine multiplying it all out.
We'd have $x$ multiplied by itself four times, which would give us $x^4$ as the highest power term.
So, the degree is 4!

STEP 5

The leading coefficient is the number multiplying the highest power of $x$ .

STEP 6

When we multiply out $A(x)$ , the term with the highest power of $x$ comes from multiplying the $x$ from each factor: $-(x)(x)(x)(x) = -x^4$ .
So, the leading coefficient is -1.

STEP 7

The zeros of a polynomial are the values of $x$ that make the polynomial equal to zero.

STEP 8

Since $A(x)$ is factored, we can see that the zeros are the values that make each factor equal to zero.
So, the zeros are $x = \mathbf{-4}$ , $x = \mathbf{-1}$ , $x = \mathbf{3}$ , and $x = \mathbf{8}$ .

STEP 9

The end behavior describes what happens to the polynomial as $x$ gets very large (positive or negative).

STEP 10

Since our leading coefficient is negative and the degree is even, both ends of the graph will point downwards.
As $x$ gets very large and negative, $A(x)$ gets very large and negative.
As $x$ gets very large and positive, $A(x)$ also gets very large and negative.

STEP 11

We know the zeros are at $-4$ , $-1$ , $3$ , and $8$ , so the graph crosses the x-axis at these points.

STEP 12

Since the degree is 4, the graph can have up to three turning points.

STEP 13

We also know the end behavior: both ends point down.

STEP 14

With this information, we can sketch a graph that passes through the zeros and has the correct end behavior.
It'll look something like a "W" flipped upside down!

STEP 15

\begin{array}{lcc} \multicolumn{3}{c}{A(x)=-(x+4)(x+1)(x-3)(x-8)} \\ \text{Degree} & 4 & \text{Zeros} \\ \text{Leading Coefficient} & -1 & -4, -1, 3, 8 \\ \end{array}
End Behavior: As $x$ gets larger and larger negative, $A(x)$ gets larger and larger negative.
As $x$ gets larger and larger positive, $A(x)$ gets larger and larger negative.
(See the sketch described in step 2.5, which should resemble an upside-down "W" passing through the zeros.)

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Studdy Solution

STEP 1

STEP 2

1. Find the Degree
2. Find the Leading Coefficient
3. Find the Zeros
4. Describe the End Behavior
5. Sketch the Graph

STEP 3

The degree of a polynomial is the highest power of $x$ when it's fully expanded.
Our polynomial $A(x) = -(x+4)(x+1)(x-3)(x-8)$ is in factored form.

STEP 4

Imagine multiplying it all out.
We'd have $x$ multiplied by itself four times, which would give us $x^4$ as the highest power term.
So, the degree is 4!

STEP 5

The leading coefficient is the number multiplying the highest power of $x$ .

STEP 6

When we multiply out $A(x)$ , the term with the highest power of $x$ comes from multiplying the $x$ from each factor: $-(x)(x)(x)(x) = -x^4$ .
So, the leading coefficient is -1.

STEP 7

The zeros of a polynomial are the values of $x$ that make the polynomial equal to zero.

STEP 8

Since $A(x)$ is factored, we can see that the zeros are the values that make each factor equal to zero.
So, the zeros are $x = \mathbf{-4}$ , $x = \mathbf{-1}$ , $x = \mathbf{3}$ , and $x = \mathbf{8}$ .

STEP 9

The end behavior describes what happens to the polynomial as $x$ gets very large (positive or negative).

STEP 10

Since our leading coefficient is negative and the degree is even, both ends of the graph will point downwards.
As $x$ gets very large and negative, $A(x)$ gets very large and negative.
As $x$ gets very large and positive, $A(x)$ also gets very large and negative.

STEP 11

We know the zeros are at $-4$ , $-1$ , $3$ , and $8$ , so the graph crosses the x-axis at these points.

STEP 12

Since the degree is 4, the graph can have up to three turning points.

STEP 13

We also know the end behavior: both ends point down.

STEP 14

With this information, we can sketch a graph that passes through the zeros and has the correct end behavior.
It'll look something like a "W" flipped upside down!

STEP 15

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Studdy Solution

STEP 1

STEP 2

1. Find the Degree2. Find the Leading Coefficient3. Find the Zeros4. Describe the End Behavior5. Sketch the Graph

STEP 3

The **degree** of a polynomial is the highest power of xxx when it's fully expanded.Our polynomial A(x)=−(x+4)(x+1)(x−3)(x−8)A(x) = -(x+4)(x+1)(x-3)(x-8)A(x)=−(x+4)(x+1)(x−3)(x−8) is in factored form.

STEP 4

Imagine multiplying it all out.We'd have xxx multiplied by itself four times, which would give us x4x^4x4 as the highest power term.So, the **degree** is **4**!

STEP 5

The **leading coefficient** is the number multiplying the highest power of xxx.

STEP 6

When we multiply out A(x)A(x)A(x), the term with the highest power of xxx comes from multiplying the xxx from each factor: −(x)(x)(x)(x)=−x4-(x)(x)(x)(x) = -x^4−(x)(x)(x)(x)=−x4.So, the **leading coefficient** is **-1**.

STEP 7

The **zeros** of a polynomial are the values of xxx that make the polynomial equal to zero.

STEP 8

Since A(x)A(x)A(x) is factored, we can see that the zeros are the values that make each factor equal to zero.So, the **zeros** are x=−4x = \mathbf{-4}x=−4, x=−1x = \mathbf{-1}x=−1, x=3x = \mathbf{3}x=3, and x=8x = \mathbf{8}x=8.

STEP 9

The **end behavior** describes what happens to the polynomial as xxx gets very large (positive or negative).

STEP 10

Since our **leading coefficient** is **negative** and the **degree** is **even**, both ends of the graph will point downwards.As xxx gets very large and negative, A(x)A(x)A(x) gets very large and negative.As xxx gets very large and positive, A(x)A(x)A(x) also gets very large and negative.

STEP 11

We know the **zeros** are at −4-4−4, −1-1−1, 333, and 888, so the graph crosses the x-axis at these points.

STEP 12

Since the **degree** is **4**, the graph can have up to three turning points.

STEP 13

We also know the **end behavior**: both ends point down.

STEP 14

With this information, we can sketch a graph that passes through the zeros and has the correct end behavior.It'll look something like a "W" flipped upside down!

STEP 15

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1. Find the Degree
2. Find the Leading Coefficient
3. Find the Zeros
4. Describe the End Behavior
5. Sketch the Graph

The degree of a polynomial is the highest power of $x$ when it's fully expanded.
Our polynomial $A(x) = -(x+4)(x+1)(x-3)(x-8)$ is in factored form.

Imagine multiplying it all out.
We'd have $x$ multiplied by itself four times, which would give us $x^4$ as the highest power term.
So, the degree is 4!

The leading coefficient is the number multiplying the highest power of $x$ .

When we multiply out $A(x)$ , the term with the highest power of $x$ comes from multiplying the $x$ from each factor: $-(x)(x)(x)(x) = -x^4$ .
So, the leading coefficient is -1.

The zeros of a polynomial are the values of $x$ that make the polynomial equal to zero.

Since $A(x)$ is factored, we can see that the zeros are the values that make each factor equal to zero.
So, the zeros are $x = \mathbf{-4}$ , $x = \mathbf{-1}$ , $x = \mathbf{3}$ , and $x = \mathbf{8}$ .

The end behavior describes what happens to the polynomial as $x$ gets very large (positive or negative).

Since our leading coefficient is negative and the degree is even, both ends of the graph will point downwards.
As $x$ gets very large and negative, $A(x)$ gets very large and negative.
As $x$ gets very large and positive, $A(x)$ also gets very large and negative.

We know the zeros are at $-4$ , $-1$ , $3$ , and $8$ , so the graph crosses the x-axis at these points.

Since the degree is 4, the graph can have up to three turning points.

We also know the end behavior: both ends point down.

With this information, we can sketch a graph that passes through the zeros and has the correct end behavior.
It'll look something like a "W" flipped upside down!