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PR4Sec4Updates_from_feedback
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@sheppardw
sheppardw committed May 25, 2024
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</objectives>
<subsection>
<title>Activities</title>
<introduction>
<remark xml:id="x-intercepts">

<statement><p> Recall that to find the <m>x</m> intercepts of a linear or quadratic function, let <m>y=0</m> and solve for <m>x</m>.</p>
</statement></remark></introduction>
<activity xml:id="Quadratic-Zeros">
<introduction>
<remark xml:id="x-intercepts">

<statement><p> Recall that to find the <m>x</m> intercepts of a linear or quadratic function, let <m>y=0</m> and solve for <m>x</m>.</p>
</statement></remark></introduction>


<task>
<statement>
<p>What is the factored form of the function, <m>f(x)=x^2+2x-15</m>?
<ol marker= "A."><li> 0</li>
<ol marker= "A.">
<li> <m>(x-3)(x+5)</m></li>
<li> <m>(x+3)(x-5)</m></li>
<li> <m>(x-3)(x-5)</m></li>
Expand All @@ -33,7 +34,7 @@
<task>
<statement>
<p>What are the <m>x</m> intercepts of the function above?
<ol marker= "A."><li> 0</li>
<ol marker= "A.">
<li> <m>-3</m> and <m>5</m></li>
<li> <m>3</m> and <m>-5</m></li>
<li> <m>-3</m> and <m>-5</m></li>
Expand All @@ -44,14 +45,14 @@
</task>
<task>
<statement>
<p> How do the <m>x</m> intercepts relate to the factored form of the function?
<p> How do the <m>x</m> intercepts relate to the factored form of the function? How can we generalize this result for polynomial functions?
</p>
</statement>
</task>

</activity>
<definition xml:id="Real-zeros">
<statement> <p>Real zeros of a polynomial function are the same as the <m>x</m> intercepts.
<statement> <p><term>Real zeros</term> of a polynomial function are the same as the <m>x</m> intercepts.
</p>
</statement>
</definition>
Expand All @@ -66,10 +67,8 @@

<activity xml:id="zeros-cubic-factoring">
<statement>
<p>Find the zeros of the polynomial by factoring <m>f(x)=x^3-x^2-56x</m>.
<ol marker= "A."><li> 0</li>
<li> <m>-7</m></li>
<li> <m>8</m></li>
<p>Find the zeros of the polynomial using the Theorem above and what we know about x intercepts. <m>f(x)=x^3-x^2-56x</m>.
<ol marker= "A.">
<li> <m>-7,8</m></li>
<li> <m>0,-7,8</m></li>
<li> <m>7,-8</m></li>
Expand All @@ -81,9 +80,9 @@
<activity xml:id="zeros-cubic-division-with-linear-factor">
<task>
<statement>
<p>Given <m>x-3</m> is a factor of the polynomial <m> f(x)=2x^3-7x^2-33x+108</m>, find the remaining factors using division.
<p>Given <m>x-3</m> is a factor of the polynomial <m> f(x)=2x^3-7x^2-33x+108</m>, find the remaining factors.
<ol marker= "A.">
<li> <m>(x+4),(2x-9)</m></li>
<li> <m>(x+4)(2x-9)</m></li>
<li> <m>(x-4)(2x+9))</m></li>
<li> <m>(2x+3)(x-9))</m></li>
<li> <m>(2x-3)(x+9))</m></li>
Expand All @@ -105,31 +104,38 @@
</task>
</activity>

<definition xml:id="multiplicity">
<statement>
<p> The <term>multiplicity</term> <idx><h>polynomial function</h><h>multiplicity</h></idx> of a zero is the number of times the corresponding linear factor appears in the factored form of the polynomial function.
</p>
</statement>
</definition>
<definition xml:id="degree-of-poly">
<statement>
<p> The <term>degree</term> <idx><h>polynomial function</h><h>multiplicity</h></idx> of a polynomial function is the sum of the multiplicities of the zeros.</p>
<p> The <term>degree</term> <idx><h>polynomial function</h><h>multiplicity</h></idx> of a polynomial function is the highest power of <m>x</m> in the expanded form of the function.</p>
</statement>
</definition>
<activity xml:id="degree-linear-example">
<statement><p>Given the function <m>5y-4=x</m>, what is the degree of the function?
<task> <statement><p>Given the function <m>5y-4=x</m>, what is the degree of the function?
<ol marker= "A.">
<li> <m>0</m></li>
<li> <m>1</m></li>
<li> <m>5</m></li>
</ol>
</p>
</statement>
</task>
<task>
<statement>
<p>Find the zeros of the linear function above. How many are there?
<ol marker= "A.">
<li> <m>0</m></li>
<li> <m>1</m></li>
<li> <m>2</m></li>
<li> <m>{1}{5}</m></li>
</ol>
</p>
</statement>
</task>
</activity>
<activity xml:id="degree-quadratic-example">
<task>
<statement>
<p>Given <m>y=5x^2+7x-6</m>, what is the degree of the function?
<p>Given the function <m>f(x)=5x^2+7x-6</m>, what is the degree of the function?
<ol marker= "A.">
<li> <m>0</m></li>
<li> <m>1</m></li>
Expand All @@ -156,28 +162,46 @@
<theorem>
<title>Fundamental Theorem of Algebra</title>
<statement>
<p> A polynomial function of degree <m>n > 0</m> has at least one zero.</p>
<p> A polynomial function of degree <m>n > 0</m> has at least one zero. Nonzero polynomials of degree <m>n</m> have exactly <m>n</m> zeros.</p>
</statement>
</theorem>

<definition xml:id="multiplicity">
<statement>
<p> The <term>multiplicity</term> <idx><h>polynomial function</h><h>multiplicity</h></idx> of a zero is the number of times the corresponding linear factor appears in the factored form of the polynomial function.
</p>
</statement>
</definition>

<activity xml:id="Degree-polynomial-factored">
<task>
<statement>
<p>Given the polynomial,<m>f(x)=(3x-2)(x+1)^2(x-6)^3</m>, find all the zeros of the function with their corresponding multiplicities. What is the degree of the function?
<ol marker= "A.">
<li> <m>3</m></li>
<li> <m>5</m></li>
<li> <m>6</m></li>
</ol>
<p>Given the polynomial,<m>f(x)=(3x-2)(x+1)^2(x-6)^3</m>, find all of the zeros of the function with their corresponding multiplicities.

</p>
</statement>
</task>
</activity>
<task>
<statement>
<p>Find the degree of the polynomial,<m>f(x)=(3x-2)(x+1)^2(x-6)^3</m>.
</p> <ol marker= "A.">
<li> <m>1</m></li>
<li> <m>2</m></li>
<li> <m>3</m></li>
<li> <m>6</m></li>
</ol>
</statement>
</task>
<task>
<statement>
<p>How do all of the exponents on the factors relate to the degree of the polynomial?</p></statement></task>
<task>
<statement>
<p>How do the multiplicities of the zeros relate to the degree of the polynomial function?</p></statement></task></activity>

<activity xml:id="Degree-polynomial-factor">
<task>
<statement>
<p>Given <m>f(x)=x^4-5x^3+x^2+21x-18</m>, find all the zeros of the function with their corresponding multiplicities. What is the degree of the function?
<p>Given <m>f(x)=x^4-5x^3+x^2+21x-18</m>, find all of the zeros of the function with their corresponding multiplicities. What is the degree of the function?
<ol marker= "A.">
<li> <m>5</m></li>
<li> <m>4</m></li>
Expand All @@ -200,20 +224,20 @@
</activity>

<activity xml:id="write-polynomial-given-zeros">
<task>

<statement>
<p>Given <m>-1</m> is a zero with multiplicity <m>2</m>, <m>4</m> is a zero with multiplicity <m>1</m> and <m>7</m> is a zero with multiplicity <m>3</m>, write a polynomial function in factored form.
</p>
</statement>
</task>

</activity>

<activity xml:id="Complex-zeros-quadratic">
<task>

<statement>
<p>Find the complex zeros for the function, <m>f(x)=x^2+16</m>.</p>
</statement>
</task>

</activity>
<theorem>
<title>Conjugate Zeros Theorem</title>
Expand All @@ -225,7 +249,7 @@

<activity xml:id="poly-graph">
<introduction>
<p>Write the polynomial function in factored form using information from the graph below.</p>
<p>Write a polynomial function in factored form using information from the graph below.</p>

<figure xml:id="fig-sage-poly-graph">
<image>
Expand Down Expand Up @@ -281,15 +305,15 @@
</task>
<task>
<statement>
<p> Write a function for the graph <m>f(x)</m> in factored form using linear factors.</p>
<p> Write a function for <m>f(x)</m> in factored form using linear factors.</p>
</statement>
</task>
</activity>
<activity xml:id="poly-create">
<introduction>
<p>The zeros of a function are <m>x=2</m>, with multiplicity <m>1</m>, <m>x=-1</m>, with multiplicity <m>2</m> and <m>x=i</m>. </p>
<p>The zeros of a function are <m>x=2</m>, with multiplicity <m>1</m>, <m>x=-1</m>, with multiplicity <m>2</m> and <m>x=i</m>, with multiplicity <m>1</m>. </p>
</introduction>
<task>
<task>
<statement>
<p> Given the information above, find a polynomial function with real coefficients of least degree. </p>
</statement>
Expand Down

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