Third-degree, With Zeros Of 3, 1, And 2, And A Y-intercept Of 7.p(x)=?
Understanding how to determine the polynomial function given specific zeros and a y-intercept is a fundamental skill in algebra. In this article, we will explore how to find a cubic polynomial \( p(x) \) with zeros at 3, 1, and 2, and a y-intercept of 7. We will break down each step involved, discuss the general form of such polynomials, and provide examples to solidify your understanding.
What is a Third-Degree Polynomial?
A third-degree polynomial, also known as a cubic polynomial, is a polynomial function where the highest exponent of the variable \( x \) is 3. The general form of a cubic polynomial is:
\[
p(x) = ax^3 + bx^2 + cx + d
\]
where \( a \neq 0 \), and \( b, c, d \) are real coefficients.
Cubic polynomials are widely used in various fields such as physics, engineering, and economics because of their versatility in modeling complex behaviors like inflection points and turning points.
Zeros of a Polynomial and Their Significance
Zeros (or roots) of a polynomial are the values of \( x \) where \( p(x) = 0 \). They are critical in understanding the shape and intercepts of the graph of the polynomial.
For a cubic polynomial, the Fundamental Theorem of Algebra states that it will have exactly three roots (considering complex roots), which could be real or complex.
In our case, the zeros are given as 3, 1, and 2, which are real roots.
Constructing the Polynomial from Zeros
Given the zeros, the polynomial can be expressed as a product of factors corresponding to each zero:
\[
p(x) = a(x - r1)(x - r2)(x - r_3)
\]
where \( r1, r2, r_3 \) are the zeros.
For zeros at 3, 1, and 2, the factors are:
\[
p(x) = a(x - 3)(x - 1)(x - 2)
\]
The constant \( a \) is a scaling factor that affects the overall shape and size of the graph.
Determining the Leading Coefficient \( a \)
To fully specify \( p(x) \), we need to find the value of \( a \). This is where the y-intercept comes into play.
The y-intercept occurs when \( x = 0 \). The value of the polynomial at \( x = 0 \), denoted as \( p(0) \), is given as 7.
So,
\[
p(0) = a(0 - 3)(0 - 1)(0 - 2) = 7
\]
Calculating the product inside:
\[
(0 - 3) = -3 \\
(0 - 1) = -1 \\
(0 - 2) = -2
\]
Multiplying these:
\[
(-3) \times (-1) \times (-2) = (-3) \times 2 = -6
\]
Therefore:
\[
p(0) = a \times (-6) = 7
\]
Solving for \( a \):
\[
a = \frac{7}{-6} = -\frac{7}{6}
\]
Thus, the polynomial function is:
\[
p(x) = -\frac{7}{6}(x - 3)(x - 1)(x - 2)
\]
Expanded Form of the Polynomial
While the factored form is useful, expanding the polynomial provides a complete algebraic expression.
Step 1: Expand two factors:
\[
(x - 3)(x - 1) = x^2 - x - 3x + 3 = x^2 - 4x + 3
\]
Step 2: Multiply the result by \( (x - 2) \):
\[
(x^2 - 4x + 3)(x - 2)
\]
Using distribution:
\[
x^2 \times x = x^3 \\
x^2 \times (-2) = -2x^2 \\
-4x \times x = -4x^2 \\
-4x \times (-2) = 8x \\
3 \times x = 3x \\
3 \times (-2) = -6
\]
Now, sum all terms:
\[
x^3 + (-2x^2) + (-4x^2) + 8x + 3x - 6
\]
Combine like terms:
\[
x^3 + (-2x^2 - 4x^2) + (8x + 3x) - 6 = x^3 - 6x^2 + 11x - 6
\]
Step 3: Incorporate the coefficient \( a = -\frac{7}{6} \):
\[
p(x) = -\frac{7}{6}(x^3 - 6x^2 + 11x - 6)
\]
Distribute:
\[
p(x) = -\frac{7}{6}x^3 + \frac{7}{6} \times 6 x^2 - \frac{7}{6} \times 11 x + \frac{7}{6} \times 6
\]
Simplify each term:
\[
p(x) = -\frac{7}{6}x^3 + 7x^2 - \frac{77}{6}x + 7
\]
This is the expanded form of the polynomial:
\[
\boxed{
p(x) = -\frac{7}{6}x^3 + 7x^2 - \frac{77}{6}x + 7
}
\]
Graphing the Polynomial and Its Features
Understanding the features of \( p(x) \) helps in visualizing its graph.
Zeros (Roots):
The polynomial crosses the x-axis at \( x = 1, 2, 3 \).
Y-intercept:
At \( x=0 \), \( p(0) = 7 \).
End Behavior:
Since the leading coefficient is negative (\( -\frac{7}{6} \)), as \( x \to \infty \), \( p(x) \to -\infty \), and as \( x \to -\infty \), \( p(x) \to \infty \).
Turning Points:
A cubic polynomial can have up to two turning points (local maxima and minima). The exact locations can be found by taking the derivative and analyzing critical points.
Applications and Importance of Understanding Such Polynomials
Knowing how to construct a polynomial from zeros and a y-intercept is essential in various mathematical and real-world contexts:
- Designing Curves: Engineers and designers use polynomial functions to create smooth curves with specific intersection points and behaviors.
- Data Modeling: Polynomial regression relies on these principles to fit data points with a curve that best represents the data.
- Physics and Economics: Modeling phenomena like projectile motion or supply-demand curves often involves cubic functions with known roots and intercepts.
- Mathematical Education: Learning to construct polynomials from roots and intercepts reinforces understanding of polynomial properties and the Fundamental Theorem of Algebra.
Summary and Key Takeaways
- A third-degree polynomial with zeros at 3, 1, and 2 can be expressed as \( p(x) = a(x - 3)(x - 1)(x - 2) \).
- To find the coefficient \( a \), use the given y-intercept \( p(0) = 7 \).
- Substituting \( x = 0 \) into the factored form allows solving for \( a \).
- The resulting polynomial in expanded form is:
\[
p(x) = -\frac{7}{6}x^3 + 7x^2 - \frac{77}{6}x + 7
\]
- Understanding how zeros, intercepts, and coefficients relate helps in graphing and analyzing polynomial functions.
By mastering these concepts, students and professionals can confidently construct and analyze cubic functions tailored to specific criteria, enhancing their problem-solving toolkit in algebra and calculus.
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Keywords: cubic polynomial, third-degree polynomial, polynomial zeros, y-intercept, algebra, polynomial expansion, graphing polynomial functions, algebraic modeling
