Given the polynomial function [tex]$f(x)=2(x+1)^2(x-2)^2(x-3)$[/tex].
i) What is the degree of [tex]$f$[/tex]?
ii) Determine the zeros of [tex]$f$[/tex] and their multiplicities.
Also, determine whether the graph of [tex]$f$[/tex] crosses or touches the [tex]$x$[/tex]-axis at each zero.
iii) Determine its [tex]$y$[/tex]-intercept.
iv) Determine the end behavior of [tex]$f$[/tex].
v) Sketch the graph of the polynomial function.
Make sure your graph shows all intercepts and exhibits the proper end behaviour.
Answer (1)
The degree of the polynomial is found by summing the exponents, resulting in 5 .
The zeros are determined by setting each factor to zero: x = − 1 (multiplicity 2), x = 2 (multiplicity 2), and x = 3 (multiplicity 1); the graph touches the x-axis at x = − 1 and x = 2 , and crosses at x = 3 .
The y-intercept is calculated by evaluating f ( 0 ) , which gives − 24 .
The end behavior is determined by the leading term 2 x 5 : as x → ∞ , f ( x ) → ∞ and as x → − ∞ , f ( x ) → − ∞ .
Explanation
Understanding the Problem We are given the polynomial function f ( x ) = 2 ( x + 1 ) 2 ( x − 2 ) 2 ( x − 3 ) . We need to find its degree, zeros and their multiplicities, whether the graph crosses or touches the x-axis at each zero, its y-intercept, end behavior, and sketch its graph.
Finding the Degree The degree of the polynomial is the sum of the exponents of each factor. In this case, the degree is 2 + 2 + 1 = 5 .
Finding the Zeros The zeros of the function are the values of x for which f ( x ) = 0 . These occur when x + 1 = 0 , x − 2 = 0 , and x − 3 = 0 . Thus, the zeros are x = − 1 , x = 2 , and x = 3 .
Determining Multiplicities The multiplicity of a zero is the exponent of its corresponding factor. The zero x = − 1 has multiplicity 2, the zero x = 2 has multiplicity 2, and the zero x = 3 has multiplicity 1.
Cross or Touch If the multiplicity of a zero is even, the graph touches the x-axis at that zero. If the multiplicity is odd, the graph crosses the x-axis at that zero. Therefore, the graph touches the x-axis at x = − 1 and x = 2 , and crosses the x-axis at x = 3 .
Finding the y-intercept The y -intercept is the value of f ( 0 ) . We have f ( 0 ) = 2 ( 0 + 1 ) 2 ( 0 − 2 ) 2 ( 0 − 3 ) = 2 ( 1 ) ( 4 ) ( − 3 ) = − 24 .
Determining End Behavior To determine the end behavior, we consider the leading term. Expanding the polynomial, the leading term is 2 x 5 . Since the degree is odd and the leading coefficient is positive, as x r i g h t a rro w in f t y , f ( x ) r i g h t a rro w in f t y and as x r i g h t a rro w − in f t y , f ( x ) r i g h t a rro w − in f t y .
Sketching the Graph Now we sketch the graph. We plot the zeros at x = − 1 , x = 2 , and x = 3 . The graph touches the x-axis at x = − 1 and x = 2 , and crosses the x-axis at x = 3 . The y -intercept is at ( 0 , − 24 ) . The end behavior is as described above.
Final Answer In summary: i) The degree of f is 5. ii) The zeros of f are x = − 1 (multiplicity 2), x = 2 (multiplicity 2), and x = 3 (multiplicity 1). The graph touches the x-axis at x = − 1 and x = 2 , and crosses the x-axis at x = 3 .
iii) The y -intercept is − 24 .
iv) As x r i g h t a rro w in f t y , f ( x ) r i g h t a rro w in f t y and as x r i g h t a rro w − in f t y , f ( x ) r i g h t a rro w − in f t y .
Examples
Polynomial functions are used to model various real-world phenomena, such as the trajectory of a projectile, the growth of a population, or the behavior of electrical circuits. Understanding the properties of polynomial functions, such as their degree, zeros, intercepts, and end behavior, allows us to make predictions and solve problems in these areas. For example, engineers use polynomial functions to design bridges and buildings, while economists use them to model market trends.
