Math / Algebra

QuestionDescribe the end behavior of a $14^{\text {th }}$ degree polynomial with a positive leading coefficient.
Describe the end behavior of a $9^{\text {th }}$ degree polynomial with a negative leading coefficient.

Studdy Solution

STEP 1

1. The degree of the polynomial determines the general shape of its graph.
2. The sign of the leading coefficient affects the direction of the ends of the graph.
3. We are considering the behavior of the polynomial as $x \to \infty$ and $x \to -\infty$ .

STEP 2

1. Analyze the end behavior of a 14th degree polynomial with a positive leading coefficient.
2. Analyze the end behavior of a 9th degree polynomial with a negative leading coefficient.

STEP 3

For a polynomial of even degree, the ends of the graph will go in the same direction.

STEP 4

Since the leading coefficient is positive, as $x \to \infty$ , the polynomial $P(x) \to \infty$ .

STEP 5

Similarly, as $x \to -\infty$ , the polynomial $P(x) \to \infty$ .

STEP 6

For a polynomial of odd degree, the ends of the graph will go in opposite directions.

STEP 7

Since the leading coefficient is negative, as $x \to \infty$ , the polynomial $P(x) \to -\infty$ .

STEP 8

As $x \to -\infty$ , the polynomial $P(x) \to \infty$ .
The end behavior of the 14th degree polynomial with a positive leading coefficient is: - As $x \to \infty$ , $P(x) \to \infty$ . - As $x \to -\infty$ , $P(x) \to \infty$ .
The end behavior of the 9th degree polynomial with a negative leading coefficient is: - As $x \to \infty$ , $P(x) \to -\infty$ . - As $x \to -\infty$ , $P(x) \to \infty$ .

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QuestionDescribe the end behavior of a $14^{\text {th }}$ degree polynomial with a positive leading coefficient.
Describe the end behavior of a $9^{\text {th }}$ degree polynomial with a negative leading coefficient.

Studdy Solution

STEP 1

1. The degree of the polynomial determines the general shape of its graph.
2. The sign of the leading coefficient affects the direction of the ends of the graph.
3. We are considering the behavior of the polynomial as $x \to \infty$ and $x \to -\infty$ .

STEP 2

1. Analyze the end behavior of a 14th degree polynomial with a positive leading coefficient.
2. Analyze the end behavior of a 9th degree polynomial with a negative leading coefficient.

STEP 3

For a polynomial of even degree, the ends of the graph will go in the same direction.

STEP 4

Since the leading coefficient is positive, as $x \to \infty$ , the polynomial $P(x) \to \infty$ .

STEP 5

Similarly, as $x \to -\infty$ , the polynomial $P(x) \to \infty$ .

STEP 6

For a polynomial of odd degree, the ends of the graph will go in opposite directions.

STEP 7

Since the leading coefficient is negative, as $x \to \infty$ , the polynomial $P(x) \to -\infty$ .

STEP 8

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QuestionDescribe the end behavior of a 14th 14^{\text {th }}14th degree polynomial with a positive leading coefficient.Describe the end behavior of a 9th 9^{\text {th }}9th degree polynomial with a negative leading coefficient.

Studdy Solution

STEP 1

1. The degree of the polynomial determines the general shape of its graph.2. The sign of the leading coefficient affects the direction of the ends of the graph.3. We are considering the behavior of the polynomial as x→∞ x \to \infty x→∞ and x→−∞ x \to -\infty x→−∞.

STEP 2

1. Analyze the end behavior of a 14th degree polynomial with a positive leading coefficient.2. Analyze the end behavior of a 9th degree polynomial with a negative leading coefficient.

STEP 3

For a polynomial of even degree, the ends of the graph will go in the same direction.

STEP 4

Since the leading coefficient is positive, as x→∞ x \to \infty x→∞, the polynomial P(x)→∞ P(x) \to \infty P(x)→∞.

STEP 5

Similarly, as x→−∞ x \to -\infty x→−∞, the polynomial P(x)→∞ P(x) \to \infty P(x)→∞.

STEP 6

For a polynomial of odd degree, the ends of the graph will go in opposite directions.

STEP 7

Since the leading coefficient is negative, as x→∞ x \to \infty x→∞, the polynomial P(x)→−∞ P(x) \to -\infty P(x)→−∞.

STEP 8

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QuestionDescribe the end behavior of a $14^{\text {th }}$ degree polynomial with a positive leading coefficient.
Describe the end behavior of a $9^{\text {th }}$ degree polynomial with a negative leading coefficient.

1. The degree of the polynomial determines the general shape of its graph.
2. The sign of the leading coefficient affects the direction of the ends of the graph.
3. We are considering the behavior of the polynomial as $x \to \infty$ and $x \to -\infty$ .

1. Analyze the end behavior of a 14th degree polynomial with a positive leading coefficient.
2. Analyze the end behavior of a 9th degree polynomial with a negative leading coefficient.

Since the leading coefficient is positive, as $x \to \infty$ , the polynomial $P(x) \to \infty$ .

Similarly, as $x \to -\infty$ , the polynomial $P(x) \to \infty$ .

Since the leading coefficient is negative, as $x \to \infty$ , the polynomial $P(x) \to -\infty$ .