Question Find the standard form, roots, degree, and leading coefficient of the function $g(x) = -x(x-1)(x-3)$ .

Studdy Solution

STEP 1

Assumptions
1. The function $g(x)$ is given in factored form as $-x(x-1)(x-3)$ .
2. We need to find the standard form of $g(x)$ .
3. We need to identify the roots of $g(x)$ .
4. We need to determine the degree of $g(x)$ .
5. We need to find the leading coefficient of $g(x)$ .

STEP 2

To convert the factored form of $g(x)$ to standard form, we need to multiply the factors together.
$g(x) = -x(x-1)(x-3)$

STEP 3

First, multiply the factors $x$ and $(x-1)$ .
$x(x-1) = x^2 - x$

STEP 4

Now, multiply the result from STEP_3 by the remaining factor $(x-3)$ .
$g(x) = -(x^2 - x)(x-3)$

STEP 5

Distribute the multiplication across the terms.
$g(x) = -(x^3 - 3x^2 - x^2 + 3x)$

STEP 6

Combine like terms within the parentheses.
$g(x) = -(x^3 - 4x^2 + 3x)$

STEP 7

Now, distribute the negative sign to each term inside the parentheses to get the standard form.
$g(x) = -x^3 + 4x^2 - 3x$

STEP 8

The standard form of $g(x)$ is:
$g(x) = -x^3 + 4x^2 - 3x$

STEP 9

To find the roots of $g(x)$ , we set $g(x)$ equal to zero and solve for $x$ .
$-x(x-1)(x-3) = 0$

STEP 10

The roots are the values of $x$ that make each factor equal to zero. Set each factor equal to zero and solve for $x$ .
$-x = 0, \quad x-1 = 0, \quad x-3 = 0$

STEP 11

Solve each equation for $x$ .
$x = 0, \quad x = 1, \quad x = 3$

STEP 12

The roots of $g(x)$ are:
$x = 0, 1, 3$

STEP 13

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

STEP 14

Looking at the standard form of $g(x)$ , we see that the highest power of $x$ is 3.
$Degree\, of\, g(x) = 3$

STEP 15

The leading coefficient of a polynomial is the coefficient of the term with the highest power of $x$ .

STEP 16

In the standard form of $g(x)$ , the coefficient of $x^3$ is -1.
$Leading\, coefficient\, of\, g(x) = -1$
Solution: a) The standard form of $g(x)$ is $-x^3 + 4x^2 - 3x$ . b) The roots of $g(x)$ are $0, 1, 3$ . c) The degree of $g(x)$ is $3$ . d) The leading coefficient of $g(x)$ is $-1$ .

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