At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Experience the convenience of getting reliable answers to your questions from a vast network of knowledgeable experts. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
Absolutely, let's explore how we would approach solving the function [tex]\( -2x^3 + 4x^2 + 5x \)[/tex].
### Understanding the Polynomial
The function given to us is:
[tex]\[ f(x) = -2x^3 + 4x^2 + 5x \][/tex]
This is a cubic polynomial in one variable [tex]\( x \)[/tex].
### Step-by-Step Solution
1. Identify the terms:
The function has three terms:
- [tex]\(-2x^3\)[/tex]: This is the cubic term, where [tex]\( a_3 = -2 \)[/tex].
- [tex]\(4x^2\)[/tex]: This is the quadratic term, where [tex]\( a_2 = 4 \)[/tex].
- [tex]\(5x\)[/tex]: This is the linear term, where [tex]\( a_1 = 5 \)[/tex].
2. Determine the degree:
The highest power of [tex]\( x \)[/tex] in the polynomial is 3. Thus, the degree of the polynomial is 3.
3. Finding roots:
To find the roots, or zeros, of the polynomial, we need to solve:
[tex]\[ -2x^3 + 4x^2 + 5x = 0 \][/tex]
Factoring out [tex]\( x \)[/tex]:
[tex]\[ x(-2x^2 + 4x + 5) = 0 \][/tex]
This gives us one root immediately:
[tex]\[ x = 0 \][/tex]
For the quadratic term, we solve:
[tex]\[ -2x^2 + 4x + 5 = 0 \][/tex]
We can use the quadratic formula for this ( [tex]\(ax^2 + bx + c = 0\)[/tex] ):
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = -2 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = 5 \)[/tex].
Calculate the discriminant:
[tex]\[ b^2 - 4ac = 4^2 - 4(-2)(5) = 16 + 40 = 56 \][/tex]
Thus, the roots are:
[tex]\[ x = \frac{-4 \pm \sqrt{56}}{-4} \][/tex]
Simplify [tex]\(\sqrt{56}\)[/tex]:
[tex]\[ \sqrt{56} = \sqrt{4 \times 14} = 2\sqrt{14} \][/tex]
Now, we substitute back in:
[tex]\[ x = \frac{-4 \pm 2\sqrt{14}}{-4} = \frac{-4 \pm 2\sqrt{14}}{-4} = 1 \mp \frac{\sqrt{14}}{2} \][/tex]
So, the other two roots are:
[tex]\[ x = 1 + \frac{\sqrt{14}}{2}, \quad x = 1 - \frac{\sqrt{14}}{2} \][/tex]
4. Graphical representation:
The general shape of the graph of [tex]\( -2x^3 + 4x^2 + 5x \)[/tex] can be understood by considering the leading term [tex]\(-2x^3\)[/tex]. Since the coefficient is negative, the graph will start from negative infinity as [tex]\( x \to -\infty \)[/tex] and will end at negative infinity as [tex]\( x \to \infty \)[/tex], producing an overall 'N' shape.
### Key Takeaways
- The polynomial [tex]\( -2x^3 + 4x^2 + 5x \)[/tex] is cubic.
- The roots found are [tex]\( 0 \)[/tex], [tex]\( 1 + \frac{\sqrt{14}}{2} \)[/tex], and [tex]\( 1 - \frac{\sqrt{14}}{2} \)[/tex].
- The shape of the graph is determined by the leading term, which is cubic and negative.
This concisely explores the function and its attributes. If you need further properties like maxima, minima, or inflection points, do let me know!
### Understanding the Polynomial
The function given to us is:
[tex]\[ f(x) = -2x^3 + 4x^2 + 5x \][/tex]
This is a cubic polynomial in one variable [tex]\( x \)[/tex].
### Step-by-Step Solution
1. Identify the terms:
The function has three terms:
- [tex]\(-2x^3\)[/tex]: This is the cubic term, where [tex]\( a_3 = -2 \)[/tex].
- [tex]\(4x^2\)[/tex]: This is the quadratic term, where [tex]\( a_2 = 4 \)[/tex].
- [tex]\(5x\)[/tex]: This is the linear term, where [tex]\( a_1 = 5 \)[/tex].
2. Determine the degree:
The highest power of [tex]\( x \)[/tex] in the polynomial is 3. Thus, the degree of the polynomial is 3.
3. Finding roots:
To find the roots, or zeros, of the polynomial, we need to solve:
[tex]\[ -2x^3 + 4x^2 + 5x = 0 \][/tex]
Factoring out [tex]\( x \)[/tex]:
[tex]\[ x(-2x^2 + 4x + 5) = 0 \][/tex]
This gives us one root immediately:
[tex]\[ x = 0 \][/tex]
For the quadratic term, we solve:
[tex]\[ -2x^2 + 4x + 5 = 0 \][/tex]
We can use the quadratic formula for this ( [tex]\(ax^2 + bx + c = 0\)[/tex] ):
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = -2 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = 5 \)[/tex].
Calculate the discriminant:
[tex]\[ b^2 - 4ac = 4^2 - 4(-2)(5) = 16 + 40 = 56 \][/tex]
Thus, the roots are:
[tex]\[ x = \frac{-4 \pm \sqrt{56}}{-4} \][/tex]
Simplify [tex]\(\sqrt{56}\)[/tex]:
[tex]\[ \sqrt{56} = \sqrt{4 \times 14} = 2\sqrt{14} \][/tex]
Now, we substitute back in:
[tex]\[ x = \frac{-4 \pm 2\sqrt{14}}{-4} = \frac{-4 \pm 2\sqrt{14}}{-4} = 1 \mp \frac{\sqrt{14}}{2} \][/tex]
So, the other two roots are:
[tex]\[ x = 1 + \frac{\sqrt{14}}{2}, \quad x = 1 - \frac{\sqrt{14}}{2} \][/tex]
4. Graphical representation:
The general shape of the graph of [tex]\( -2x^3 + 4x^2 + 5x \)[/tex] can be understood by considering the leading term [tex]\(-2x^3\)[/tex]. Since the coefficient is negative, the graph will start from negative infinity as [tex]\( x \to -\infty \)[/tex] and will end at negative infinity as [tex]\( x \to \infty \)[/tex], producing an overall 'N' shape.
### Key Takeaways
- The polynomial [tex]\( -2x^3 + 4x^2 + 5x \)[/tex] is cubic.
- The roots found are [tex]\( 0 \)[/tex], [tex]\( 1 + \frac{\sqrt{14}}{2} \)[/tex], and [tex]\( 1 - \frac{\sqrt{14}}{2} \)[/tex].
- The shape of the graph is determined by the leading term, which is cubic and negative.
This concisely explores the function and its attributes. If you need further properties like maxima, minima, or inflection points, do let me know!
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
