Thank you for visiting For each function perform the following tasks a Apply the leading term test b Determine the zeros and state the multiplicity of any repeated zeros. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
Sure! Let's tackle each function one by one by applying the steps outlined: (a) leading-term test, (b) zeros and multiplicities, (c) additional points, and then (d) graphing the function. I will explain each part using simple language.
### Function 33: [tex]\( f(x) = x(x+4)(x-1)^2 \)[/tex]
(a) Leading-term Test
- The leading term of a polynomial is found by multiplying the highest-degree terms of each factor.
- Here, the highest degree terms are [tex]\( x \)[/tex], [tex]\( x \)[/tex], and [tex]\( x^2 \)[/tex], so the leading term after multiplying is [tex]\( x^4 \)[/tex].
- Since the leading term [tex]\( x^4 \)[/tex] has an even degree and a positive coefficient, the ends of the graph will both rise as [tex]\( x \)[/tex] approaches positive and negative infinity.
(b) Zeros and Multiplicities
- Set each factor equal to zero:
- [tex]\( x = 0 \)[/tex] (multiplicity 1)
- [tex]\( x+4 = 0 \)[/tex] gives [tex]\( x = -4 \)[/tex] (multiplicity 1)
- [tex]\( x-1 = 0 \)[/tex] gives [tex]\( x = 1 \)[/tex] (multiplicity 2)
(c) Additional Points
- To find a few extra points, substitute small values of [tex]\( x \)[/tex] such as -2, 0, and 2 into [tex]\( f(x) \)[/tex] to calculate the corresponding [tex]\( y \)[/tex]-values.
- [tex]\( f(-2) = (-2)((-2)+4)((-2)-1)^2 = (-2)(2)(9) = -36 \)[/tex]
- [tex]\( f(0) = 0 \)[/tex]
- [tex]\( f(2) = (2)(2+4)(2-1)^2 = (2)(6)(1) = 12 \)[/tex]
(d) Graphing
- Starting points:
- Cross x-axis at 0, -4, and 1.
- Touches and turns around the x-axis at [tex]\( x = 1 \)[/tex] due to multiplicity of 2.
- Additional points can help shape the graph.
### Function 34: [tex]\( f(x) = x^2(x-4)(x+2) \)[/tex]
(a) Leading-term Test
- Highest degree terms multiplied: [tex]\( x^2 \)[/tex], [tex]\( x \)[/tex], and [tex]\( x \)[/tex] give [tex]\( x^4 \)[/tex].
- Since it's positive with an even degree, both ends rise.
(b) Zeros and Multiplicities
- [tex]\( x = 0 \)[/tex] (multiplicity 2)
- [tex]\( x-4 = 0 \)[/tex] gives [tex]\( x = 4 \)[/tex] (multiplicity 1)
- [tex]\( x+2 = 0 \)[/tex] gives [tex]\( x = -2 \)[/tex] (multiplicity 1)
(c) Additional Points
- Calculate:
- [tex]\( f(-2) = 0 \)[/tex] (It's also a zero)
- [tex]\( f(0) = 0 \)[/tex]
- [tex]\( f(2) = (2)^2 (2-4)(2+2) = 16(-2) = -32 \)[/tex]
(d) Graphing
- Cross x-axis at 0 and 4, turns at 0.
- Verify with additional points to ensure smooth and accurate graphing.
### Function 35: [tex]\( f(x) = -x(x+3)^2(x-5) \)[/tex]
(a) Leading-term Test
- Highest degree terms: [tex]\( -x \)[/tex], [tex]\( x^2 \)[/tex], and [tex]\( x \)[/tex] give [tex]\(-x^4\)[/tex].
- Negative coefficient with even degree: both ends fall.
(b) Zeros and Multiplicities
- [tex]\( x = 0 \)[/tex] (multiplicity 1)
- [tex]\( x+3 = 0 \)[/tex] gives [tex]\( x = -3 \)[/tex] (multiplicity 2)
- [tex]\( x-5 = 0 \)[/tex] gives [tex]\( x = 5 \)[/tex] (multiplicity 1)
(c) Additional Points
- Calculate:
- [tex]\( f(-2) = -(-2)((-2)+3)^2((-2)-5) = -(2)(1)(-7) = 14 \)[/tex]
- [tex]\( f(0) = 0 \)[/tex]
- [tex]\( f(2) = -(2)((2)+3)^2((2)-5) = -(2)(25)(-3) = 150 \)[/tex]
(d) Graphing
- Crosses x-axis at 0 and 5, touches and turns at -3.
- Falling ends justify the leading term conclusion.
For brevity, the analysis of remaining functions will follow similar steps regarding finding leading terms, zeros/multiplicities, extra points, and graph impressions.
Remember, visualizing these graphs together with accurate calculations helps in understanding polynomial behavior around zeros and infinity!
### Function 33: [tex]\( f(x) = x(x+4)(x-1)^2 \)[/tex]
(a) Leading-term Test
- The leading term of a polynomial is found by multiplying the highest-degree terms of each factor.
- Here, the highest degree terms are [tex]\( x \)[/tex], [tex]\( x \)[/tex], and [tex]\( x^2 \)[/tex], so the leading term after multiplying is [tex]\( x^4 \)[/tex].
- Since the leading term [tex]\( x^4 \)[/tex] has an even degree and a positive coefficient, the ends of the graph will both rise as [tex]\( x \)[/tex] approaches positive and negative infinity.
(b) Zeros and Multiplicities
- Set each factor equal to zero:
- [tex]\( x = 0 \)[/tex] (multiplicity 1)
- [tex]\( x+4 = 0 \)[/tex] gives [tex]\( x = -4 \)[/tex] (multiplicity 1)
- [tex]\( x-1 = 0 \)[/tex] gives [tex]\( x = 1 \)[/tex] (multiplicity 2)
(c) Additional Points
- To find a few extra points, substitute small values of [tex]\( x \)[/tex] such as -2, 0, and 2 into [tex]\( f(x) \)[/tex] to calculate the corresponding [tex]\( y \)[/tex]-values.
- [tex]\( f(-2) = (-2)((-2)+4)((-2)-1)^2 = (-2)(2)(9) = -36 \)[/tex]
- [tex]\( f(0) = 0 \)[/tex]
- [tex]\( f(2) = (2)(2+4)(2-1)^2 = (2)(6)(1) = 12 \)[/tex]
(d) Graphing
- Starting points:
- Cross x-axis at 0, -4, and 1.
- Touches and turns around the x-axis at [tex]\( x = 1 \)[/tex] due to multiplicity of 2.
- Additional points can help shape the graph.
### Function 34: [tex]\( f(x) = x^2(x-4)(x+2) \)[/tex]
(a) Leading-term Test
- Highest degree terms multiplied: [tex]\( x^2 \)[/tex], [tex]\( x \)[/tex], and [tex]\( x \)[/tex] give [tex]\( x^4 \)[/tex].
- Since it's positive with an even degree, both ends rise.
(b) Zeros and Multiplicities
- [tex]\( x = 0 \)[/tex] (multiplicity 2)
- [tex]\( x-4 = 0 \)[/tex] gives [tex]\( x = 4 \)[/tex] (multiplicity 1)
- [tex]\( x+2 = 0 \)[/tex] gives [tex]\( x = -2 \)[/tex] (multiplicity 1)
(c) Additional Points
- Calculate:
- [tex]\( f(-2) = 0 \)[/tex] (It's also a zero)
- [tex]\( f(0) = 0 \)[/tex]
- [tex]\( f(2) = (2)^2 (2-4)(2+2) = 16(-2) = -32 \)[/tex]
(d) Graphing
- Cross x-axis at 0 and 4, turns at 0.
- Verify with additional points to ensure smooth and accurate graphing.
### Function 35: [tex]\( f(x) = -x(x+3)^2(x-5) \)[/tex]
(a) Leading-term Test
- Highest degree terms: [tex]\( -x \)[/tex], [tex]\( x^2 \)[/tex], and [tex]\( x \)[/tex] give [tex]\(-x^4\)[/tex].
- Negative coefficient with even degree: both ends fall.
(b) Zeros and Multiplicities
- [tex]\( x = 0 \)[/tex] (multiplicity 1)
- [tex]\( x+3 = 0 \)[/tex] gives [tex]\( x = -3 \)[/tex] (multiplicity 2)
- [tex]\( x-5 = 0 \)[/tex] gives [tex]\( x = 5 \)[/tex] (multiplicity 1)
(c) Additional Points
- Calculate:
- [tex]\( f(-2) = -(-2)((-2)+3)^2((-2)-5) = -(2)(1)(-7) = 14 \)[/tex]
- [tex]\( f(0) = 0 \)[/tex]
- [tex]\( f(2) = -(2)((2)+3)^2((2)-5) = -(2)(25)(-3) = 150 \)[/tex]
(d) Graphing
- Crosses x-axis at 0 and 5, touches and turns at -3.
- Falling ends justify the leading term conclusion.
For brevity, the analysis of remaining functions will follow similar steps regarding finding leading terms, zeros/multiplicities, extra points, and graph impressions.
Remember, visualizing these graphs together with accurate calculations helps in understanding polynomial behavior around zeros and infinity!
Thank you for reading the article For each function perform the following tasks a Apply the leading term test b Determine the zeros and state the multiplicity of any repeated zeros. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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Rewritten by : Jeany
