root_linear_coefficient_theorem.yaml

author_notes: ''
auto_tts_enabled: true
blurb: ''
category: Algebra
correctness_feedback_enabled: false
edits_allowed: true
init_state_name: Start
language_code: en
objective: discover the Root Linear Coefficient Theorem
param_changes: []
param_specs: {}
schema_version: 47
states:
  Coefficient Signs:
    classifier_model_id: null
    content:
      content_id: content
      html: "<p>Here's that expression again:</p><p><oppia-noninteractive-math math_content-with-value=\"\
        {&amp;quot;raw_latex&amp;quot;: &amp;quot;(x - a_1)(x - a_2)(x - a_3)...(x\
        \ - a_n)&amp;quot;, &amp;quot;svg_filename&amp;quot;: &amp;quot;mathImg_20200904_135011_dpupkuy4ou_height_2d731_width_36d59_vertical_0d833.svg&amp;quot;}\"\
        ></oppia-noninteractive-math><br><br>If\_<oppia-noninteractive-math math_content-with-value=\"\
        {&amp;quot;raw_latex&amp;quot;: &amp;quot;n&amp;quot;, &amp;quot;svg_filename&amp;quot;:\
        \ &amp;quot;mathImg_20200904_131209_bn2zj4i2te_height_1d429_width_1d405_vertical_0d241.svg&amp;quot;}\"\
        ></oppia-noninteractive-math>\_is even, what's the sign of the constant term?\
        \ \_How about the linear term?</p><p><br>Likewise, if\_<oppia-noninteractive-math\
        \ math_content-with-value=\"{&amp;quot;raw_latex&amp;quot;: &amp;quot;n&amp;quot;,\
        \ &amp;quot;svg_filename&amp;quot;: &amp;quot;mathImg_20200904_131209_bn2zj4i2te_height_1d429_width_1d405_vertical_0d241.svg&amp;quot;}\"\
        ></oppia-noninteractive-math>\_is odd, what's the sign of the constant term?\
        \ \_What's the sign of the linear term?</p>"
    interaction:
      answer_groups:
      - outcome:
          dest: Quotient of terms
          feedback:
            content_id: feedback_1
            html: <p>Good.  As a side note, what do you think the sign is of the x^i
              term given n is even? Given n is odd?</p>
          labelled_as_correct: false
          missing_prerequisite_skill_id: null
          param_changes: []
          refresher_exploration_id: null
        rule_specs:
        - inputs:
            x: 0
          rule_type: Equals
        tagged_skill_misconception_id: null
        training_data: []
      - outcome:
          dest: Coefficient signs WA1
          feedback:
            content_id: feedback_2
            html: <p>Hmm... that's a bit weird isn't it?  The expression had only
              negative signs, and your answer has all positive signs...</p>
          labelled_as_correct: false
          missing_prerequisite_skill_id: null
          param_changes: []
          refresher_exploration_id: null
        rule_specs:
        - inputs:
            x: 1
          rule_type: Equals
        tagged_skill_misconception_id: null
        training_data: []
      - outcome:
          dest: Coefficient signs WA1
          feedback:
            content_id: feedback_3
            html: <p>That looks correct, but let's try a sample case just to make
              sure.</p>
          labelled_as_correct: false
          missing_prerequisite_skill_id: null
          param_changes: []
          refresher_exploration_id: null
        rule_specs:
        - inputs:
            x: 2
          rule_type: Equals
        tagged_skill_misconception_id: null
        training_data: []
      - outcome:
          dest: Coefficient signs WA2
          feedback:
            content_id: feedback_4
            html: <p>That looks a little fishy... let's try a sample case just to
              be sure...</p>
          labelled_as_correct: false
          missing_prerequisite_skill_id: null
          param_changes: []
          refresher_exploration_id: null
        rule_specs:
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          rule_type: Equals
        tagged_skill_misconception_id: null
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            html: '<p><oppia-noninteractive-math math_content-with-value="{&amp;quot;raw_latex&amp;quot;:
              &amp;quot;+,-,-,+&amp;quot;, &amp;quot;svg_filename&amp;quot;: &amp;quot;mathImg_20200904_134604_kjwad50w4g_height_2d258_width_10d408_vertical_0d715.svg&amp;quot;}"></oppia-noninteractive-math></p>'
          - content_id: ca_choices_6
            html: '<p><oppia-noninteractive-math math_content-with-value="{&amp;quot;raw_latex&amp;quot;:
              &amp;quot;+,+,+,+&amp;quot;, &amp;quot;svg_filename&amp;quot;: &amp;quot;mathImg_20200904_134658_e5dg8ij01b_height_2d258_width_10d408_vertical_0d715.svg&amp;quot;}"></oppia-noninteractive-math></p>'
          - content_id: ca_choices_7
            html: '<p><oppia-noninteractive-math math_content-with-value="{&amp;quot;raw_latex&amp;quot;:
              &amp;quot;+,-,+,-&amp;quot;, &amp;quot;svg_filename&amp;quot;: &amp;quot;mathImg_20200904_134757_f1uwdf95b2_height_2d258_width_10d408_vertical_0d715.svg&amp;quot;}"></oppia-noninteractive-math></p>'
          - content_id: ca_choices_8
            html: '<p><oppia-noninteractive-math math_content-with-value="{&amp;quot;raw_latex&amp;quot;:
              &amp;quot;-,+,-,+&amp;quot;, &amp;quot;svg_filename&amp;quot;: &amp;quot;mathImg_20200904_134842_r40fdf6mk0_height_2d258_width_10d408_vertical_0d715.svg&amp;quot;}"></oppia-noninteractive-math></p>'
        showChoicesInShuffledOrder:
          value: false
      default_outcome:
        dest: Coefficient Signs
        feedback:
          content_id: default_outcome
          html: <p>Oops, if you have reached this case, something has gone wrong...</p>
        labelled_as_correct: false
        missing_prerequisite_skill_id: null
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      hints: []
      id: MultipleChoiceInput
      solution: null
    next_content_id_index: 9
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    recorded_voiceovers:
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        feedback_2: {}
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        feedback_3: {}
        feedback_4: {}
  Coefficient signs WA1:
    classifier_model_id: null
    content:
      content_id: content
      html: "<p>If we tried a simple case, like:<br><br></p><p><oppia-noninteractive-math\
        \ math_content-with-value=\"{&amp;quot;raw_latex&amp;quot;: &amp;quot;(x -\
        \ 3)(x - 4)(x - 1)&amp;quot;, &amp;quot;svg_filename&amp;quot;: &amp;quot;mathImg_20200904_145032_kpinnt0x2d_height_2d731_width_21d582_vertical_0d833.svg&amp;quot;}\"\
        ></oppia-noninteractive-math></p><p>\_</p><p>where\_<oppia-noninteractive-math\
        \ math_content-with-value=\"{&amp;quot;raw_latex&amp;quot;: &amp;quot;a_1\
        \ = 3&amp;quot;, &amp;quot;svg_filename&amp;quot;: &amp;quot;mathImg_20200904_145112_zdet2aoua9_height_2d376_width_6d584_vertical_0d596.svg&amp;quot;}\"\
        ></oppia-noninteractive-math>,\_<oppia-noninteractive-math math_content-with-value=\"\
        {&amp;quot;raw_latex&amp;quot;: &amp;quot;a_2=4&amp;quot;, &amp;quot;svg_filename&amp;quot;:\
        \ &amp;quot;mathImg_20200904_145208_5bmpx75x83_height_2d376_width_6d584_vertical_0d596.svg&amp;quot;}\"\
        ></oppia-noninteractive-math>, and\_<oppia-noninteractive-math math_content-with-value=\"\
        {&amp;quot;raw_latex&amp;quot;: &amp;quot;a_3 = 1&amp;quot;, &amp;quot;svg_filename&amp;quot;:\
        \ &amp;quot;mathImg_20200904_145257_o9m38za85u_height_2d376_width_6d584_vertical_0d596.svg&amp;quot;}\"\
        ></oppia-noninteractive-math>, then you're suggesting that the constant term\
        \ should be positive, right?</p><p>\_</p><p>What is the constant term?</p>"
    interaction:
      answer_groups:
      - outcome:
          dest: Coefficient Signs
          feedback:
            content_id: feedback_1
            html: <p>Right!  But wait-- that's not positive.  Let's try the previous
              question again...</p>
          labelled_as_correct: false
          missing_prerequisite_skill_id: null
          param_changes: []
          refresher_exploration_id: null
        rule_specs:
        - inputs:
            x: -12.0
          rule_type: Equals
        tagged_skill_misconception_id: null
        training_data: []
      confirmed_unclassified_answers: []
      customization_args: {}
      default_outcome:
        dest: Coefficient signs WA1
        feedback:
          content_id: default_outcome
          html: <p>Try again.  We want the constant term.</p>
        labelled_as_correct: false
        missing_prerequisite_skill_id: null
        param_changes: []
        refresher_exploration_id: null
      hints: []
      id: NumericInput
      solution: null
    next_content_id_index: 2
    param_changes: []
    recorded_voiceovers:
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        content: {}
        default_outcome: {}
        feedback_1: {}
  Coefficient signs WA2:
    classifier_model_id: null
    content:
      content_id: content
      html: "<p>You said \"-,+,-,+\" meaning if you expand:</p><p><oppia-noninteractive-math\
        \ math_content-with-value=\"{&amp;quot;raw_latex&amp;quot;: &amp;quot;(x -\
        \ a_1) (x - a_2) (x - a_3) ... (x - a_n)&amp;quot;, &amp;quot;svg_filename&amp;quot;:\
        \ &amp;quot;mathImg_20200904_135011_dpupkuy4ou_height_2d731_width_36d59_vertical_0d833.svg&amp;quot;}\"\
        ></oppia-noninteractive-math>,<br>and\_<oppia-noninteractive-math math_content-with-value=\"\
        {&amp;quot;raw_latex&amp;quot;: &amp;quot;n&amp;quot;, &amp;quot;svg_filename&amp;quot;:\
        \ &amp;quot;mathImg_20200904_131209_bn2zj4i2te_height_1d429_width_1d405_vertical_0d241.svg&amp;quot;}\"\
        ></oppia-noninteractive-math>\_is even, you should get that the constant term\
        \ is negative. \_Let's try it on this sample case:</p><p>\_</p><p><oppia-noninteractive-math\
        \ math_content-with-value=\"{&amp;quot;raw_latex&amp;quot;: &amp;quot;(x -\
        \ 3) (x - 4)&amp;quot;, &amp;quot;svg_filename&amp;quot;: &amp;quot;mathImg_20200904_145457_e8wewljydb_height_2d731_width_14d388_vertical_0d833.svg&amp;quot;}\"\
        ></oppia-noninteractive-math><br><br>What is the coefficient of the constant\
        \ term?</p>"
    interaction:
      answer_groups:
      - outcome:
          dest: Coefficient Signs
          feedback:
            content_id: feedback_1
            html: <p>Right.  And that's not negative.  Let's try the previous question
              again...</p>
          labelled_as_correct: false
          missing_prerequisite_skill_id: null
          param_changes: []
          refresher_exploration_id: null
        rule_specs:
        - inputs:
            x: 12.0
          rule_type: Equals
        tagged_skill_misconception_id: null
        training_data: []
      confirmed_unclassified_answers: []
      customization_args: {}
      default_outcome:
        dest: Coefficient signs WA2
        feedback:
          content_id: default_outcome
          html: <p>Nope!  Try again.</p>
        labelled_as_correct: false
        missing_prerequisite_skill_id: null
        param_changes: []
        refresher_exploration_id: null
      hints: []
      id: NumericInput
      solution: null
    next_content_id_index: 2
    param_changes: []
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        feedback_1: {}
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        feedback_1: {}
  END:
    classifier_model_id: null
    content:
      content_id: content
      html: <p>Congratulations, you have finished!</p>
    interaction:
      answer_groups: []
      confirmed_unclassified_answers: []
      customization_args:
        recommendedExplorationIds:
          value: []
      default_outcome: null
      hints: []
      id: EndExploration
      solution: null
    next_content_id_index: 0
    param_changes: []
    recorded_voiceovers:
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  More General Expansion:
    classifier_model_id: null
    content:
      content_id: content
      html: "<p>What about\_<oppia-noninteractive-math math_content-with-value=\"\
        {&amp;quot;raw_latex&amp;quot;: &amp;quot;(x-a)(x-b)(x-c)&amp;quot;, &amp;quot;svg_filename&amp;quot;:\
        \ &amp;quot;mathImg_20200904_150030_q2zopa1qyw_height_2d731_width_21d329_vertical_0d833.svg&amp;quot;}\"\
        ></oppia-noninteractive-math>?</p>"
    interaction:
      answer_groups:
      - outcome:
          dest: Particular coefficients (part 1)
          feedback:
            content_id: feedback_1
            html: <p>Great!</p>
          labelled_as_correct: false
          missing_prerequisite_skill_id: null
          param_changes: []
          refresher_exploration_id: null
        rule_specs:
        - inputs:
            x: 2
          rule_type: Equals
        tagged_skill_misconception_id: null
        training_data: []
      confirmed_unclassified_answers: []
      customization_args:
        choices:
          value:
          - content_id: ca_choices_2
            html: '<p><oppia-noninteractive-math math_content-with-value="{&amp;quot;raw_latex&amp;quot;:
              &amp;quot;x^3 - a x^2 - b x - c&amp;quot;, &amp;quot;svg_filename&amp;quot;:
              &amp;quot;mathImg_20200904_145644_k7kgyhp35x_height_2d494_width_17d962_vertical_0d36.svg&amp;quot;}"></oppia-noninteractive-math></p>'
          - content_id: ca_choices_3
            html: '<p><oppia-noninteractive-math math_content-with-value="{&amp;quot;raw_latex&amp;quot;:
              &amp;quot;x^3 + (a+b+c)x^2 + (ab+bc+ca)x + abc&amp;quot;, &amp;quot;svg_filename&amp;quot;:
              &amp;quot;mathImg_20200904_145731_sug9iaaiaj_height_2d968_width_42d822_vertical_0d833.svg&amp;quot;}"></oppia-noninteractive-math></p>'
          - content_id: ca_choices_4
            html: '<p><oppia-noninteractive-math math_content-with-value="{&amp;quot;raw_latex&amp;quot;:
              &amp;quot;x^3 - (a+b+c)x^2 + (ab+bc+ca)x - abc&amp;quot;, &amp;quot;svg_filename&amp;quot;:
              &amp;quot;mathImg_20200904_145820_opah350bca_height_2d968_width_42d822_vertical_0d833.svg&amp;quot;}"></oppia-noninteractive-math></p>'
          - content_id: ca_choices_5
            html: '<p><oppia-noninteractive-math math_content-with-value="{&amp;quot;raw_latex&amp;quot;:
              &amp;quot;x^3 + (a+b+c)x^2 - (ab+bc+ca)x + abc&amp;quot;, &amp;quot;svg_filename&amp;quot;:
              &amp;quot;mathImg_20200904_145933_tv24zk1jjf_height_2d968_width_42d822_vertical_0d833.svg&amp;quot;}"></oppia-noninteractive-math></p>'
        showChoicesInShuffledOrder:
          value: false
      default_outcome:
        dest: More General Expansion
        feedback:
          content_id: default_outcome
          html: <p>Nope!  Try again.</p>
        labelled_as_correct: false
        missing_prerequisite_skill_id: null
        param_changes: []
        refresher_exploration_id: null
      hints: []
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        feedback_1: {}
  Particular coefficients (part 1):
    classifier_model_id: null
    content:
      content_id: content
      html: "<p>Did you notice something about the previous answer? \_What happens\
        \ if we consider an even more general expression:<br><oppia-noninteractive-math\
        \ math_content-with-value=\"{&amp;quot;raw_latex&amp;quot;: &amp;quot;(x -\
        \ a_1)(x - a_2)(x - a_3)...(x - a_n)&amp;quot;, &amp;quot;svg_filename&amp;quot;:\
        \ &amp;quot;mathImg_20200904_135011_dpupkuy4ou_height_2d731_width_36d59_vertical_0d833.svg&amp;quot;}\"\
        ></oppia-noninteractive-math><br><br>What is the absolute value of the constant\
        \ term (the term with no x's)?</p>"
    interaction:
      answer_groups:
      - outcome:
          dest: Particular coefficients (part 2)
          feedback:
            content_id: feedback_1
            html: <p>That's correct!</p>
          labelled_as_correct: false
          missing_prerequisite_skill_id: null
          param_changes: []
          refresher_exploration_id: null
        rule_specs:
        - inputs:
            x: 3
          rule_type: Equals
        tagged_skill_misconception_id: null
        training_data: []
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              &amp;quot;a_1&amp;quot;, &amp;quot;svg_filename&amp;quot;: &amp;quot;mathImg_20200904_133655_ou7e9pxilh_height_1d784_width_2d302_vertical_0d596.svg&amp;quot;}"></oppia-noninteractive-math></p>'
          - content_id: ca_choices_3
            html: '<p><oppia-noninteractive-math math_content-with-value="{&amp;quot;raw_latex&amp;quot;:
              &amp;quot;a_1 + a_2 + a_3 + ... + a_n&amp;quot;, &amp;quot;svg_filename&amp;quot;:
              &amp;quot;mathImg_20200904_144222_qonc2d8qgh_height_2d139_width_21d844_vertical_0d596.svg&amp;quot;}"></oppia-noninteractive-math></p>'
          - content_id: ca_choices_4
            html: '<p><oppia-noninteractive-math math_content-with-value="{&amp;quot;raw_latex&amp;quot;:
              &amp;quot;1&amp;quot;, &amp;quot;svg_filename&amp;quot;: &amp;quot;mathImg_20200904_132540_jbva8gbc53_height_2d021_width_1d173_vertical_0d241.svg&amp;quot;}"></oppia-noninteractive-math></p>'
          - content_id: ca_choices_5
            html: '<p><oppia-noninteractive-math math_content-with-value="{&amp;quot;raw_latex&amp;quot;:
              &amp;quot;a_1 a_2 a_3 ... a_n&amp;quot;, &amp;quot;svg_filename&amp;quot;:
              &amp;quot;mathImg_20200904_144401_1b7zx4oho0_height_1d784_width_12d505_vertical_0d596.svg&amp;quot;}"></oppia-noninteractive-math></p>'
        showChoicesInShuffledOrder:
          value: false
      default_outcome:
        dest: Particular coefficients (part 1)
        feedback:
          content_id: default_outcome
          html: <p>Nope!  Try again.</p>
        labelled_as_correct: false
        missing_prerequisite_skill_id: null
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        refresher_exploration_id: null
      hints: []
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  Particular coefficients (part 2):
    classifier_model_id: null
    content:
      content_id: content
      html: "<p>Again, consider the expression:<br><oppia-noninteractive-math math_content-with-value=\"\
        {&amp;quot;raw_latex&amp;quot;: &amp;quot;(x - a_1)(x - a_2)(x - a_3)...(x\
        \ - a_n)&amp;quot;, &amp;quot;svg_filename&amp;quot;: &amp;quot;mathImg_20200904_135011_dpupkuy4ou_height_2d731_width_36d59_vertical_0d833.svg&amp;quot;}\"\
        ></oppia-noninteractive-math><br><br>What is the absolute value of the linear\
        \ term (the\_<oppia-noninteractive-math math_content-with-value=\"{&amp;quot;raw_latex&amp;quot;:\
        \ &amp;quot;x^1&amp;quot;, &amp;quot;svg_filename&amp;quot;: &amp;quot;mathImg_20200904_144914_08fk62lk4x_height_2d376_width_2d402_vertical_0d241.svg&amp;quot;}\"\
        ></oppia-noninteractive-math>\_term)?<br></p>"
    interaction:
      answer_groups:
      - outcome:
          dest: Coefficient Signs
          feedback:
            content_id: feedback_1
            html: <p>Yep!  Admit it, you just chose the longest answer...</p>
          labelled_as_correct: false
          missing_prerequisite_skill_id: null
          param_changes: []
          refresher_exploration_id: null
        rule_specs:
        - inputs:
            x: 3
          rule_type: Equals
        tagged_skill_misconception_id: null
        training_data: []
      confirmed_unclassified_answers: []
      customization_args:
        choices:
          value:
          - content_id: ca_choices_2
            html: '<p><oppia-noninteractive-math math_content-with-value="{&amp;quot;raw_latex&amp;quot;:
              &amp;quot;a_1 a_2 + a_2 a_3 + ... + a_n a_1&amp;quot;, &amp;quot;svg_filename&amp;quot;:
              &amp;quot;mathImg_20200904_144605_6v3h6dslb9_height_2d139_width_23d597_vertical_0d596.svg&amp;quot;}"></oppia-noninteractive-math></p>'
          - content_id: ca_choices_3
            html: '<p><oppia-noninteractive-math math_content-with-value="{&amp;quot;raw_latex&amp;quot;:
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  Quotient of terms:
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      html: "<p>Here's the general expression again:</p><p><oppia-noninteractive-math\
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        {&amp;quot;raw_latex&amp;quot;: &amp;quot;n&amp;quot;, &amp;quot;svg_filename&amp;quot;:\
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        ></oppia-noninteractive-math>\_is even, the constant and linear terms have\
        \ opposite signs...</p><p><br>when\_<oppia-noninteractive-math math_content-with-value=\"\
        {&amp;quot;raw_latex&amp;quot;: &amp;quot;n&amp;quot;, &amp;quot;svg_filename&amp;quot;:\
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        ></oppia-noninteractive-math>\_is odd, the constant and linear terms also\
        \ have opposite signs...<br><br>What, then, is the sign of the quotient:<br>[linear\
        \ coefficient] / [constant term]<br>?</p>"
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            html: <p>Yep.  For example, 5 / (-3) is negative, and -3 / 5 is negative,
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  Quotient part 2:
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      html: "<p>Now for the fun part. \_Do you remember how the linear coefficient\
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        ></oppia-noninteractive-math><br><br>and the constant term was:</p><p><oppia-noninteractive-math\
        \ math_content-with-value=\"{&amp;quot;raw_latex&amp;quot;: &amp;quot;(-1)^n\
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        ></oppia-noninteractive-math><br><br>(where, as discussed before, the sign\
        \ is added in).<br><br>What happens when we divide the linear coefficient\
        \ by the constant term?</p>"
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            html: '<p><oppia-noninteractive-math math_content-with-value="{&amp;quot;raw_latex&amp;quot;:
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          html: <p>Oops, that's not right.  Careful; signs can be tricky!</p>
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  Simple Expansion:
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      html: "<p>What about something a little harder? \_Can you tell me what the terms\
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        ></oppia-noninteractive-math>,\_<oppia-noninteractive-math math_content-with-value=\"\
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        ></oppia-noninteractive-math>, respectively, format your answer as the string:\
        \ a,b,c,d (without spaces).</p>"
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            html: <p>Good!  Now let's look at a slightly more general expansion.</p>
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        dest: Simple Expansion
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          html: <p>Sorry, try again.  Remember to format it exactly as shown.</p>
        labelled_as_correct: false
        missing_prerequisite_skill_id: null
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  Start:
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      html: "<p>Do you remember FOIL? \_It goes something like this:\_<oppia-noninteractive-math\
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        ></oppia-noninteractive-math>.<br></p>"
    interaction:
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      hints: []
      id: Continue
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  Theorem:
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      html: "<p>The Root Linear Coefficient Theorem is...<br><br>Given the polynomial\_\
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        ></oppia-noninteractive-math>,\_<oppia-noninteractive-math math_content-with-value=\"\
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        ></oppia-noninteractive-math>:<br><br>The sum of the reciprocals of the roots:\_\
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        ></oppia-noninteractive-math><br>is equal to the negative coefficient of the\
        \ linear term:\_<oppia-noninteractive-math math_content-with-value=\"{&amp;quot;raw_latex&amp;quot;:\
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        ></oppia-noninteractive-math><br><br>or...</p><p><br><oppia-noninteractive-math\
        \ math_content-with-value=\"{&amp;quot;raw_latex&amp;quot;: &amp;quot;c_1\
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        ></oppia-noninteractive-math></p>"
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          html: <p>Congrats!  We may use this theorem later on to prove something
            even more interesting...</p>
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states_schema_version: 42
tags: []
title: Root Linear Coefficient Theorem