Answer :
Sure, let's analyze the expression [tex]\( V = 2x^3 - 19x^2 + 39x \)[/tex].
### Step 1: Factoring the Expression
First, we want to factor the polynomial expression. Factoring can help us find the roots of the equation or simplify the expression for further calculations.
The given polynomial can be factored as:
[tex]\[ V = x(x - 3)(2x - 13) \][/tex]
This means that the expression [tex]\( 2x^3 - 19x^2 + 39x \)[/tex] can be written as the product of three linear factors.
### Step 2: Verifying the Factors
To check that the factorization is correct, you can expand [tex]\( x(x - 3)(2x - 13) \)[/tex] and see if it simplifies back to the original expression:
1. [tex]\( x(x - 3) = x^2 - 3x \)[/tex]
2. Expanding [tex]\( (x^2 - 3x)(2x - 13) \)[/tex]:
- [tex]\( x^2 \times 2x = 2x^3 \)[/tex]
- [tex]\( x^2 \times (-13) = -13x^2 \)[/tex]
- [tex]\(-3x \times 2x = -6x^2 \)[/tex]
- [tex]\(-3x \times (-13) = 39x \)[/tex]
Combine these terms:
- [tex]\( 2x^3 \)[/tex]
- [tex]\((-13x^2 - 6x^2 = -19x^2)\)[/tex]
- [tex]\(39x\)[/tex]
This confirms the factorization as [tex]\( V = x(x - 3)(2x - 13) \)[/tex].
### Step 3: Finding the Derivative
To find the rate of change of this expression, we need to compute its derivative:
The derivative of [tex]\( V = 2x^3 - 19x^2 + 39x \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[ V' = 6x^2 - 38x + 39 \][/tex]
This derivative tells us how the value of [tex]\( V \)[/tex] changes with small changes in [tex]\( x \)[/tex].
### Summary
- We factored the expression as [tex]\( x(x - 3)(2x - 13) \)[/tex].
- The derivative of the expression is [tex]\( 6x^2 - 38x + 39 \)[/tex].
This step-by-step breakdown should help you understand how the polynomial is analyzed and manipulated. If you have further questions, feel free to ask!
### Step 1: Factoring the Expression
First, we want to factor the polynomial expression. Factoring can help us find the roots of the equation or simplify the expression for further calculations.
The given polynomial can be factored as:
[tex]\[ V = x(x - 3)(2x - 13) \][/tex]
This means that the expression [tex]\( 2x^3 - 19x^2 + 39x \)[/tex] can be written as the product of three linear factors.
### Step 2: Verifying the Factors
To check that the factorization is correct, you can expand [tex]\( x(x - 3)(2x - 13) \)[/tex] and see if it simplifies back to the original expression:
1. [tex]\( x(x - 3) = x^2 - 3x \)[/tex]
2. Expanding [tex]\( (x^2 - 3x)(2x - 13) \)[/tex]:
- [tex]\( x^2 \times 2x = 2x^3 \)[/tex]
- [tex]\( x^2 \times (-13) = -13x^2 \)[/tex]
- [tex]\(-3x \times 2x = -6x^2 \)[/tex]
- [tex]\(-3x \times (-13) = 39x \)[/tex]
Combine these terms:
- [tex]\( 2x^3 \)[/tex]
- [tex]\((-13x^2 - 6x^2 = -19x^2)\)[/tex]
- [tex]\(39x\)[/tex]
This confirms the factorization as [tex]\( V = x(x - 3)(2x - 13) \)[/tex].
### Step 3: Finding the Derivative
To find the rate of change of this expression, we need to compute its derivative:
The derivative of [tex]\( V = 2x^3 - 19x^2 + 39x \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[ V' = 6x^2 - 38x + 39 \][/tex]
This derivative tells us how the value of [tex]\( V \)[/tex] changes with small changes in [tex]\( x \)[/tex].
### Summary
- We factored the expression as [tex]\( x(x - 3)(2x - 13) \)[/tex].
- The derivative of the expression is [tex]\( 6x^2 - 38x + 39 \)[/tex].
This step-by-step breakdown should help you understand how the polynomial is analyzed and manipulated. If you have further questions, feel free to ask!
