1. Write your name, date, and block on this paper. 2. Write your equation on this paper. 3. State the degree of your function. 4. State the number of total roots of your function.... 1. Write your name, date, and block on this paper. 2. Write your equation on this paper. 3. State the degree of your function. 4. State the number of total roots of your function. 5. Graph your equation on DESMOS. State the number of Real and Complex roots of your function. 6. Find the y-intercept. 7. Use DESMOS to approximate points of Relative Maximum and Relative Minimum. 8. List any Real Integer Roots and the factor that relates to them. Divide out the Real Integer Roots.
Understand the Problem
The question outlines a series of tasks related to analyzing a polynomial function. It involves writing the function, determining its properties such as degree, roots, and y-intercept, and using DESMOS for graphing and finding relative extrema.
Answer
The polynomial is $f(x) = x^3 - 21x^2 - 10x + 12$, degree $3$, total roots $3$, y-intercept $(0, 12)$.
Answer for screen readers
- Polynomial Function: $f(x) = x^3 - 21x^2 - 10x + 12$
- Degree: $3$
- Total Number of Roots: $3$
- y-Intercept: $(0, 12)$
Steps to Solve
- Write the polynomial function
The given polynomial function is:
$$ f(x) = x^3 - 21x^2 - 10x + 12 $$
- State the degree of the polynomial
The degree of the polynomial is determined by the highest exponent of the variable $x$.
In this case, the highest exponent is $3$, so:
Degree = $3$.
- Determine the total number of roots
According to the Fundamental Theorem of Algebra, the total number of roots (real and complex) of a polynomial function is equal to its degree.
Therefore, the total number of roots is $3$.
- Graph the equation on DESMOS
To graph the equation, input:
$$ f(x) = x^3 - 21x^2 - 10x + 12 $$
Using DESMOS, observe the graph for the real and complex roots.
- Find the y-intercept
The y-intercept can be determined by evaluating the function at $x = 0$:
$$ f(0) = (0)^3 - 21(0)^2 - 10(0) + 12 = 12 $$
The y-intercept is at the point $(0, 12)$.
- Approximate points of Relative Maximum and Relative Minimum
Using DESMOS, find and list the values of $x$ where the function has relative maximum and minimum values. Use the graph feature to identify these points.
- List Real Integer Roots and related factors
To find the integer roots, use the Rational Root Theorem, which suggests testing the factors of the constant term ($12$) over the leading coefficient ($1$).
Possible factors of $12$: $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$.
Test these values in the function to determine any real integer roots.
- Polynomial Function: $f(x) = x^3 - 21x^2 - 10x + 12$
- Degree: $3$
- Total Number of Roots: $3$
- y-Intercept: $(0, 12)$
More Information
The polynomial is cubic, implying it can have up to three roots. The y-intercept can often help identify where the graph intersects the y-axis. Graphing software like DESMOS will facilitate understanding the nature of roots, whether they are real or complex.
Tips
- Overlooking complex roots when counting total roots.
- Error in substituting to find the y-intercept; ensure to evaluate correctly at $x = 0$.
- Misidentifying relative extrema; ensure to analyze the graph carefully.
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