Solved on Jan 29, 2024

Determine if the degree of the polynomial $f(x) = (x-2)(x-4)(x+1)$ is odd, even, or neither.

STEP 1

Assumptions
1. The function given is $f(x)=(x-2)(x-4)(x+1)$ .
2. We need to determine if the degree of this polynomial function is odd or even.

STEP 2

The degree of a polynomial function is the highest power of $x$ when the polynomial is expressed in its standard form.

STEP 3

To find the degree of the function $f(x)$ , we need to expand the product $(x-2)(x-4)(x+1)$ .

STEP 4

First, expand the product of the first two factors $(x-2)(x-4)$ .
$(x-2)(x-4) = x^2 - 4x - 2x + 8$

STEP 5

Combine like terms in the expansion.
$(x-2)(x-4) = x^2 - 6x + 8$

STEP 6

Now, multiply this result by the third factor $(x+1)$ .
$(x^2 - 6x + 8)(x+1)$

STEP 7

Distribute each term of the first polynomial over the second polynomial.
$(x^2 - 6x + 8)(x+1) = x^3 + x^2 - 6x^2 - 6x + 8x + 8$

STEP 8

Combine like terms in the expansion.
$x^3 + x^2 - 6x^2 - 6x + 8x + 8 = x^3 - 5x^2 + 2x + 8$

STEP 9

Now that we have the expanded form of the polynomial, we can identify the degree of the polynomial.

STEP 10

The degree of the polynomial is the highest power of $x$ in the expanded form.
$\text{Degree of } f(x) = 3$

STEP 11

Since the degree of the polynomial is 3, which is an odd number, the degree of the function $f(x)$ is odd.
The degree of the function $f(x)=(x-2)(x-4)(x+1)$ is odd.

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Determine if the degree of the polynomial $f(x) = (x-2)(x-4)(x+1)$ is odd, even, or neither.

STEP 1

Assumptions
1. The function given is $f(x)=(x-2)(x-4)(x+1)$ .
2. We need to determine if the degree of this polynomial function is odd or even.

STEP 2

The degree of a polynomial function is the highest power of $x$ when the polynomial is expressed in its standard form.

STEP 3

To find the degree of the function $f(x)$ , we need to expand the product $(x-2)(x-4)(x+1)$ .

STEP 4

First, expand the product of the first two factors $(x-2)(x-4)$ .
$(x-2)(x-4) = x^2 - 4x - 2x + 8$

STEP 5

Combine like terms in the expansion.
$(x-2)(x-4) = x^2 - 6x + 8$

STEP 6

Now, multiply this result by the third factor $(x+1)$ .
$(x^2 - 6x + 8)(x+1)$

STEP 7

Distribute each term of the first polynomial over the second polynomial.
$(x^2 - 6x + 8)(x+1) = x^3 + x^2 - 6x^2 - 6x + 8x + 8$

STEP 8

Combine like terms in the expansion.
$x^3 + x^2 - 6x^2 - 6x + 8x + 8 = x^3 - 5x^2 + 2x + 8$

STEP 9

Now that we have the expanded form of the polynomial, we can identify the degree of the polynomial.

STEP 10

The degree of the polynomial is the highest power of $x$ in the expanded form.
$\text{Degree of } f(x) = 3$

STEP 11

Since the degree of the polynomial is 3, which is an odd number, the degree of the function $f(x)$ is odd.
The degree of the function $f(x)=(x-2)(x-4)(x+1)$ is odd.

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Determine if the degree of the polynomial f(x)=(x−2)(x−4)(x+1)f(x) = (x-2)(x-4)(x+1)f(x)=(x−2)(x−4)(x+1) is odd, even, or neither.

STEP 1

Assumptions1. The function given is f(x)=(x−2)(x−4)(x+1)f(x)=(x-2)(x-4)(x+1)f(x)=(x−2)(x−4)(x+1).2. We need to determine if the degree of this polynomial function is odd or even.

STEP 2

The degree of a polynomial function is the highest power of xxx when the polynomial is expressed in its standard form.

STEP 3

To find the degree of the function f(x)f(x)f(x), we need to expand the product (x−2)(x−4)(x+1)(x-2)(x-4)(x+1)(x−2)(x−4)(x+1).

STEP 4

First, expand the product of the first two factors (x−2)(x−4)(x-2)(x-4)(x−2)(x−4).(x−2)(x−4)=x2−4x−2x+8 (x-2)(x-4) = x^2 - 4x - 2x + 8 (x−2)(x−4)=x2−4x−2x+8

STEP 5

Combine like terms in the expansion.(x−2)(x−4)=x2−6x+8 (x-2)(x-4) = x^2 - 6x + 8 (x−2)(x−4)=x2−6x+8

STEP 6

Now, multiply this result by the third factor (x+1)(x+1)(x+1).(x2−6x+8)(x+1) (x^2 - 6x + 8)(x+1) (x2−6x+8)(x+1)

STEP 7

Distribute each term of the first polynomial over the second polynomial.(x2−6x+8)(x+1)=x3+x2−6x2−6x+8x+8 (x^2 - 6x + 8)(x+1) = x^3 + x^2 - 6x^2 - 6x + 8x + 8 (x2−6x+8)(x+1)=x3+x2−6x2−6x+8x+8

STEP 8

Combine like terms in the expansion.x3+x2−6x2−6x+8x+8=x3−5x2+2x+8 x^3 + x^2 - 6x^2 - 6x + 8x + 8 = x^3 - 5x^2 + 2x + 8 x3+x2−6x2−6x+8x+8=x3−5x2+2x+8

STEP 9

Now that we have the expanded form of the polynomial, we can identify the degree of the polynomial.

STEP 10

The degree of the polynomial is the highest power of xxx in the expanded form.Degree of f(x)=3 \text{Degree of } f(x) = 3 Degree of f(x)=3

STEP 11

Since the degree of the polynomial is 3, which is an odd number, the degree of the function f(x)f(x)f(x) is odd.The degree of the function f(x)=(x−2)(x−4)(x+1)f(x)=(x-2)(x-4)(x+1)f(x)=(x−2)(x−4)(x+1) is odd.

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Determine if the degree of the polynomial $f(x) = (x-2)(x-4)(x+1)$ is odd, even, or neither.

Assumptions
1. The function given is $f(x)=(x-2)(x-4)(x+1)$ .
2. We need to determine if the degree of this polynomial function is odd or even.

The degree of a polynomial function is the highest power of $x$ when the polynomial is expressed in its standard form.

To find the degree of the function $f(x)$ , we need to expand the product $(x-2)(x-4)(x+1)$ .

First, expand the product of the first two factors $(x-2)(x-4)$ .
$(x-2)(x-4) = x^2 - 4x - 2x + 8$

Combine like terms in the expansion.
$(x-2)(x-4) = x^2 - 6x + 8$

Now, multiply this result by the third factor $(x+1)$ .
$(x^2 - 6x + 8)(x+1)$

Distribute each term of the first polynomial over the second polynomial.
$(x^2 - 6x + 8)(x+1) = x^3 + x^2 - 6x^2 - 6x + 8x + 8$

Combine like terms in the expansion.
$x^3 + x^2 - 6x^2 - 6x + 8x + 8 = x^3 - 5x^2 + 2x + 8$

The degree of the polynomial is the highest power of $x$ in the expanded form.
$\text{Degree of } f(x) = 3$

Since the degree of the polynomial is 3, which is an odd number, the degree of the function $f(x)$ is odd.
The degree of the function $f(x)=(x-2)(x-4)(x+1)$ is odd.