Math / Algebra

QuestionWhat is the least possible degree of the polynomial graphed above? $\square$

Studdy Solution

STEP 1

1. The graph of the polynomial has three turning points.
2. The polynomial crosses the x-axis at three distinct points: $x = -2$ , $x = 1$ , and $x = 4$ .
3. The polynomial is a real-valued function.

STEP 2

1. Understand the relationship between turning points and the degree of a polynomial.
2. Use the x-intercepts to infer the degree of the polynomial.
3. Determine the least possible degree of the polynomial.

STEP 3

The number of turning points in a polynomial graph is at most $n-1$ , where $n$ is the degree of the polynomial.
Given that there are three turning points, the degree $n$ must satisfy $n-1 \geq 3$ .

STEP 4

The polynomial crosses the x-axis at three distinct points, which suggests that the polynomial has at least three distinct linear factors.
A polynomial with $k$ distinct roots can have a degree of at least $k$ .

STEP 5

Combining the information from the turning points and the x-intercepts, the least degree $n$ must satisfy both conditions: $n-1 \geq 3$ and $n \geq 3$ .
The smallest integer $n$ that satisfies both conditions is $n = 4$ .
The least possible degree of the polynomial is:
$\boxed{4}$

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QuestionWhat is the least possible degree of the polynomial graphed above? $\square$

Studdy Solution

STEP 1

1. The graph of the polynomial has three turning points.
2. The polynomial crosses the x-axis at three distinct points: $x = -2$ , $x = 1$ , and $x = 4$ .
3. The polynomial is a real-valued function.

STEP 2

1. Understand the relationship between turning points and the degree of a polynomial.
2. Use the x-intercepts to infer the degree of the polynomial.
3. Determine the least possible degree of the polynomial.

STEP 3

The number of turning points in a polynomial graph is at most $n-1$ , where $n$ is the degree of the polynomial.
Given that there are three turning points, the degree $n$ must satisfy $n-1 \geq 3$ .

STEP 4

The polynomial crosses the x-axis at three distinct points, which suggests that the polynomial has at least three distinct linear factors.
A polynomial with $k$ distinct roots can have a degree of at least $k$ .

STEP 5

Combining the information from the turning points and the x-intercepts, the least degree $n$ must satisfy both conditions: $n-1 \geq 3$ and $n \geq 3$ .
The smallest integer $n$ that satisfies both conditions is $n = 4$ .
The least possible degree of the polynomial is:
$\boxed{4}$

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QuestionWhat is the least possible degree of the polynomial graphed above? □\square□

Studdy Solution

STEP 1

1. The graph of the polynomial has three turning points.2. The polynomial crosses the x-axis at three distinct points: x=−2 x = -2 x=−2, x=1 x = 1 x=1, and x=4 x = 4 x=4.3. The polynomial is a real-valued function.

STEP 2

1. Understand the relationship between turning points and the degree of a polynomial.2. Use the x-intercepts to infer the degree of the polynomial.3. Determine the least possible degree of the polynomial.

STEP 3

The number of turning points in a polynomial graph is at most n−1 n-1 n−1, where n n n is the degree of the polynomial.Given that there are three turning points, the degree n n n must satisfy n−1≥3 n-1 \geq 3 n−1≥3.

STEP 4

The polynomial crosses the x-axis at three distinct points, which suggests that the polynomial has at least three distinct linear factors.A polynomial with k k k distinct roots can have a degree of at least k k k.

STEP 5

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionWhat is the least possible degree of the polynomial graphed above? $\square$

1. The graph of the polynomial has three turning points.
2. The polynomial crosses the x-axis at three distinct points: $x = -2$ , $x = 1$ , and $x = 4$ .
3. The polynomial is a real-valued function.

1. Understand the relationship between turning points and the degree of a polynomial.
2. Use the x-intercepts to infer the degree of the polynomial.
3. Determine the least possible degree of the polynomial.

The number of turning points in a polynomial graph is at most $n-1$ , where $n$ is the degree of the polynomial.
Given that there are three turning points, the degree $n$ must satisfy $n-1 \geq 3$ .

The polynomial crosses the x-axis at three distinct points, which suggests that the polynomial has at least three distinct linear factors.
A polynomial with $k$ distinct roots can have a degree of at least $k$ .