Solved on Oct 23, 2023

The degree of the polynomial $f(x) = -2x^3(x-1)(x+5)$ is $\square$ . The leading coefficient is $\square$ .

STEP 1

Assumptions1. The polynomial function is $f(x)=- x^{3}(x-1)(x+5)$ . The degree of a polynomial is the highest power of the variable in the polynomial3. The leading coefficient is the coefficient of the term with the highest degree

STEP 2

First, we need to expand the polynomial to identify the term with the highest degree.
$f(x)=-2 x^{}(x-1)(x+5)$

STEP 3

Expand the polynomial by multiplying the terms.
$f(x)=-2 x^{3} \cdot x \cdot (x-1) \cdot (x+5)$

STEP 4

Continue to expand the polynomial.
$f(x)=-2 x^{4} \cdot (x-1) \cdot (x+)$

STEP 5

Continue to expand the polynomial.
$f(x)=-2 x^{5} \cdot (x+5) -2 x^{4} \cdot (x+5)$

STEP 6

Continue to expand the polynomial.
$f(x)=-2 x^{6} +10 x^{5} -2 x^{5} +10 x^{4}$

STEP 7

implify the polynomial by combining like terms.
$f(x)=-2 x^{6} + x^{5} +10 x^{4}$

STEP 8

Now that we have the expanded polynomial, we can identify the term with the highest degree. This term will determine the degree of the polynomial.
$Degree =6$

STEP 9

The leading coefficient is the coefficient of the term with the highest degree.
$Leading\, coefficient = -2$ The degree of the polynomial function $f(x)=-2 x^{3}(x-)(x+5)$ is $6$ . The leading coefficient is $-2$ .

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The degree of the polynomial $f(x) = -2x^3(x-1)(x+5)$ is $\square$ . The leading coefficient is $\square$ .

STEP 1

Assumptions1. The polynomial function is $f(x)=- x^{3}(x-1)(x+5)$ . The degree of a polynomial is the highest power of the variable in the polynomial3. The leading coefficient is the coefficient of the term with the highest degree

STEP 2

First, we need to expand the polynomial to identify the term with the highest degree.
$f(x)=-2 x^{}(x-1)(x+5)$

STEP 3

Expand the polynomial by multiplying the terms.
$f(x)=-2 x^{3} \cdot x \cdot (x-1) \cdot (x+5)$

STEP 4

Continue to expand the polynomial.
$f(x)=-2 x^{4} \cdot (x-1) \cdot (x+)$

STEP 5

Continue to expand the polynomial.
$f(x)=-2 x^{5} \cdot (x+5) -2 x^{4} \cdot (x+5)$

STEP 6

Continue to expand the polynomial.
$f(x)=-2 x^{6} +10 x^{5} -2 x^{5} +10 x^{4}$

STEP 7

implify the polynomial by combining like terms.
$f(x)=-2 x^{6} + x^{5} +10 x^{4}$

STEP 8

Now that we have the expanded polynomial, we can identify the term with the highest degree. This term will determine the degree of the polynomial.
$Degree =6$

STEP 9

The leading coefficient is the coefficient of the term with the highest degree.
$Leading\, coefficient = -2$ The degree of the polynomial function $f(x)=-2 x^{3}(x-)(x+5)$ is $6$ . The leading coefficient is $-2$ .

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The degree of the polynomial f(x)=−2x3(x−1)(x+5)f(x) = -2x^3(x-1)(x+5)f(x)=−2x3(x−1)(x+5) is □\square□. The leading coefficient is □\square□.

STEP 1

Assumptions1. The polynomial function is f(x)=−x3(x−1)(x+5)f(x)=- x^{3}(x-1)(x+5)f(x)=−x3(x−1)(x+5). The degree of a polynomial is the highest power of the variable in the polynomial3. The leading coefficient is the coefficient of the term with the highest degree

STEP 2

First, we need to expand the polynomial to identify the term with the highest degree.f(x)=−2x(x−1)(x+5)f(x)=-2 x^{}(x-1)(x+5)f(x)=−2x(x−1)(x+5)

STEP 3

Expand the polynomial by multiplying the terms.f(x)=−2x3⋅x⋅(x−1)⋅(x+5)f(x)=-2 x^{3} \cdot x \cdot (x-1) \cdot (x+5)f(x)=−2x3⋅x⋅(x−1)⋅(x+5)

STEP 4

Continue to expand the polynomial.f(x)=−2x4⋅(x−1)⋅(x+)f(x)=-2 x^{4} \cdot (x-1) \cdot (x+)f(x)=−2x4⋅(x−1)⋅(x+)

STEP 5

Continue to expand the polynomial.f(x)=−2x5⋅(x+5)−2x4⋅(x+5)f(x)=-2 x^{5} \cdot (x+5) -2 x^{4} \cdot (x+5)f(x)=−2x5⋅(x+5)−2x4⋅(x+5)

STEP 6

Continue to expand the polynomial.f(x)=−2x6+10x5−2x5+10x4f(x)=-2 x^{6} +10 x^{5} -2 x^{5} +10 x^{4}f(x)=−2x6+10x5−2x5+10x4

STEP 7

implify the polynomial by combining like terms.f(x)=−2x6+x5+10x4f(x)=-2 x^{6} + x^{5} +10 x^{4}f(x)=−2x6+x5+10x4

STEP 8

Now that we have the expanded polynomial, we can identify the term with the highest degree. This term will determine the degree of the polynomial.Degree=6Degree =6Degree=6

STEP 9

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The degree of the polynomial $f(x) = -2x^3(x-1)(x+5)$ is $\square$ . The leading coefficient is $\square$ .

Assumptions1. The polynomial function is $f(x)=- x^{3}(x-1)(x+5)$ . The degree of a polynomial is the highest power of the variable in the polynomial3. The leading coefficient is the coefficient of the term with the highest degree

First, we need to expand the polynomial to identify the term with the highest degree.
$f(x)=-2 x^{}(x-1)(x+5)$

Expand the polynomial by multiplying the terms.
$f(x)=-2 x^{3} \cdot x \cdot (x-1) \cdot (x+5)$

Continue to expand the polynomial.
$f(x)=-2 x^{4} \cdot (x-1) \cdot (x+)$

Continue to expand the polynomial.
$f(x)=-2 x^{5} \cdot (x+5) -2 x^{4} \cdot (x+5)$

Continue to expand the polynomial.
$f(x)=-2 x^{6} +10 x^{5} -2 x^{5} +10 x^{4}$

implify the polynomial by combining like terms.
$f(x)=-2 x^{6} + x^{5} +10 x^{4}$

Now that we have the expanded polynomial, we can identify the term with the highest degree. This term will determine the degree of the polynomial.
$Degree =6$