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Magazine Chapter 5: Problem 93
Graph each polynomial function. $$ f(x)=x^{3}+2 x^{2}-5 x-6 $$
Short Answer
Expert verified
Graph has zeros at \(-3\), \(-1\), and \(2\), and y-intercept at \(-6\).
Step by step solution
01
- Identify Key Features
Examine the polynomial function and identify key features such as the degree, leading coefficient, and constant term. The function is given by \(f(x) = x^3 + 2x^2 - 5x - 6\). It is a cubic polynomial.
02
- Determine End Behavior
Use the leading term to determine the end behavior. Since it is a cubic polynomial with a positive leading coefficient, as \(x \to \infty\), \(f(x) \to \infty\), and as \(x \to -\infty\), \(f(x) \to -\infty\).
03
- Find the Zeros and Factorize
Find the zeros of the polynomial to determine the x-intercepts. This can involve factoring or using the Rational Root Theorem. The polynomial can be factored as \((x+3)(x+1)(x-2)\), yielding zeros at \(x = -3\), \(x = -1\), and \(x = 2\).
04
- Calculate the Y-Intercept
The y-intercept is found by evaluating \(f(0)\). Substituting 0 in place of \(x\) gives \(f(0) = -6\). So, the y-intercept is \( (0, -6) \).
05
- Plot Key Points
Plot the zeros \( (-3,0) \), \( (-1,0) \), \( (2,0) \), and the y-intercept \((0, -6)\) on the graph.
06
- Sketch the Graph
Using the plotted points, the end behavior, and the nature of a cubic polynomial (which will cross the x-axis at each zero), sketch the graph smoothly connecting these points, ensuring that the graph correctly represents the behavior derived in the previous steps.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cubic Polynomial
A cubic polynomial is a type of polynomial where the highest power of the variable is three. The general form is given by: \[ f(x) = ax^3 + bx^2 + cx + d \] In the given exercise, the cubic polynomial is: \[ f(x) = x^3 + 2x^2 - 5x - 6 \] Here, the coefficients are:
- a: 1 (leading coefficient, associated with \(x^3\))
- b: 2 (associated with \(x^2\))
- c: -5 (associated with \(x\))
- d: -6 (constant term)
Zeros of Polynomials
The zeros of a polynomial are the values of \(x\) that make the polynomial equal to zero. These are also the points where the graph of the polynomial crosses the x-axis. To find the zeros, we solve the equation: \[ x^3 + 2x^2 - 5x - 6 = 0 \] In this case, factoring the polynomial helps: \[ (x+3)(x+1)(x-2) = 0 \] From this factorization, we can find the zeros:
- x = -3
- x = -1
- x = 2
End Behavior
The end behavior of a polynomial describes how the values of \(f(x)\) behave as \(x\) approaches positive or negative infinity. For the given cubic polynomial \(f(x) = x^3 + 2x^2 - 5x - 6\), we look at the term with the highest power to determine end behavior. Since the leading coefficient (the coefficient of \(x^3\)) is positive:
- As \( x \to \infty \), \( f(x) \to \infty \)
- As \( x \to -\infty \), \( f(x) \to -\infty \)
Y-Intercept
The y-intercept of a polynomial is where the graph crosses the y-axis. This occurs when \(x = 0\). To find the y-intercept, we substitute \(x = 0\) into the polynomial function: \[ f(0) = 0^3 + 2(0)^2 - 5(0) - 6 = -6 \] Therefore, the y-intercept is at the point \((0, -6)\). This point is important when sketching the polynomial graph, as it provides a specific starting point on the y-axis where the graph cuts across. Plotting this alongside zeros gives a clear idea of how the polynomial behaves near the origin.
