Answer :
To analyze the polynomial expression [tex]\(4x^3 - 20x^2 + 9x - 45\)[/tex], we can follow several steps. Here’s a step-by-step breakdown:
1. Identify the Coefficients:
The polynomial given is:
[tex]\[
4x^3 - 20x^2 + 9x - 45
\][/tex]
The coefficients are:
- [tex]\(a_3 = 4\)[/tex] (coefficient of [tex]\(x^3\)[/tex])
- [tex]\(a_2 = -20\)[/tex] (coefficient of [tex]\(x^2\)[/tex])
- [tex]\(a_1 = 9\)[/tex] (coefficient of [tex]\(x\)[/tex])
- [tex]\(a_0 = -45\)[/tex] (constant term)
2. Degree of the Polynomial:
The highest power of [tex]\(x\)[/tex] in the polynomial is 3. Therefore, the degree of this polynomial is 3.
3. Factoring the Polynomial (If Needed):
To factorize a polynomial, we often look for its roots first. For a cubic polynomial, we can use techniques like synthetic division, Rational Root Theorem, or numerical methods to find its roots.
4. Roots of the Polynomial (If Solving for x):
While finding exact roots can be complex, we typically substitute values into the polynomial to check for roots or use algebraic solutions.
5. Graphical Interpretation:
A polynomial of degree 3 will resemble an “S”-shaped curve, intersecting the x-axis at its real roots. Between these roots, the curve can change direction.
6. Behavior at Infinity:
For large values of [tex]\(|x|\)[/tex], the term [tex]\(4x^3\)[/tex] will dominate and determine the behavior of the polynomial:
- As [tex]\(x \to \infty\)[/tex], [tex]\(4x^3 \to \infty\)[/tex]
- As [tex]\(x \to -\infty\)[/tex], [tex]\(4x^3 \to -\infty\)[/tex]
Here is a summarizing step:
- Calculating at Specific Points: We can substitute different values of [tex]\(x\)[/tex] into the polynomial to get a sense of its behavior. For instance:
- At [tex]\(x = 0\)[/tex]:
[tex]\[
4(0)^3 - 20(0)^2 + 9(0) - 45 = -45
\][/tex]
- At [tex]\(x = 1\)[/tex]:
[tex]\[
4(1)^3 - 20(1)^2 + 9(1) - 45 = 4 - 20 + 9 - 45 = -52
\][/tex]
By examining these points, one can better understand the polynomial’s behavior numerically and graphically.
In summary, the polynomial [tex]\(4x^3 - 20x^2 + 9x - 45\)[/tex] is a third-degree polynomial with coefficients determined as stated above. It exhibits typical behavior of a cubic function with roots and turning points, representative of an “S”-shaped curve.
1. Identify the Coefficients:
The polynomial given is:
[tex]\[
4x^3 - 20x^2 + 9x - 45
\][/tex]
The coefficients are:
- [tex]\(a_3 = 4\)[/tex] (coefficient of [tex]\(x^3\)[/tex])
- [tex]\(a_2 = -20\)[/tex] (coefficient of [tex]\(x^2\)[/tex])
- [tex]\(a_1 = 9\)[/tex] (coefficient of [tex]\(x\)[/tex])
- [tex]\(a_0 = -45\)[/tex] (constant term)
2. Degree of the Polynomial:
The highest power of [tex]\(x\)[/tex] in the polynomial is 3. Therefore, the degree of this polynomial is 3.
3. Factoring the Polynomial (If Needed):
To factorize a polynomial, we often look for its roots first. For a cubic polynomial, we can use techniques like synthetic division, Rational Root Theorem, or numerical methods to find its roots.
4. Roots of the Polynomial (If Solving for x):
While finding exact roots can be complex, we typically substitute values into the polynomial to check for roots or use algebraic solutions.
5. Graphical Interpretation:
A polynomial of degree 3 will resemble an “S”-shaped curve, intersecting the x-axis at its real roots. Between these roots, the curve can change direction.
6. Behavior at Infinity:
For large values of [tex]\(|x|\)[/tex], the term [tex]\(4x^3\)[/tex] will dominate and determine the behavior of the polynomial:
- As [tex]\(x \to \infty\)[/tex], [tex]\(4x^3 \to \infty\)[/tex]
- As [tex]\(x \to -\infty\)[/tex], [tex]\(4x^3 \to -\infty\)[/tex]
Here is a summarizing step:
- Calculating at Specific Points: We can substitute different values of [tex]\(x\)[/tex] into the polynomial to get a sense of its behavior. For instance:
- At [tex]\(x = 0\)[/tex]:
[tex]\[
4(0)^3 - 20(0)^2 + 9(0) - 45 = -45
\][/tex]
- At [tex]\(x = 1\)[/tex]:
[tex]\[
4(1)^3 - 20(1)^2 + 9(1) - 45 = 4 - 20 + 9 - 45 = -52
\][/tex]
By examining these points, one can better understand the polynomial’s behavior numerically and graphically.
In summary, the polynomial [tex]\(4x^3 - 20x^2 + 9x - 45\)[/tex] is a third-degree polynomial with coefficients determined as stated above. It exhibits typical behavior of a cubic function with roots and turning points, representative of an “S”-shaped curve.
