I had dinner. Then, a ginger tea made by my mother. Helped my father with the decluttering of the call log on his phone. Maths class. The workbook had problems on Polynomials such as:
Simplifying this expression:
49a^2+70ab+25b^2
This expression is a Perfect Square Trinomial. It follows the algebraic identity:
1. Identify the Squares
First, notice that the first and last terms are perfect squares:
* First term: 49a^2 = (7a)^2
* Last term: 25b^2 = (5b)^2
2. Verify the Middle Term
Check if the middle term (70ab) matches the 2xy part of the identity:
* Since the middle term matches perfectly, we can apply the identity.
3. The Factorized Form
Using the values x = 7a and y = 5b:
Final Answer: (7a + 5b)(7a + 5b) or simply (7a + 5b)^2.
Determining if the expression below can be factorises:
To factorize the quadratic polynomial 12x^2 + 7x + 3, we look for two numbers that multiply to a \times c and add up to b.
1. Identify the Coefficients
For a quadratic in the form ax^2 + bx + c:
* * * 2. The Product-Sum Test
We need two numbers that:
* Multiply to: 12 \times 3 = 36
* Add up to: 7
Let’s look at the factors of 36:
* 1 \times 36 (Sum: 37)
* 2 \times 18 (Sum: 20)
* 3 \times 12 (Sum: 15)
* 4 \times 9 (Sum: 13)
* 6 \times 6 (Sum: 12)
3. Conclusion on Factorization
Notice that none of the pairs of factors of 36 add up to 7. Since there are no integers that satisfy both conditions, this polynomial cannot be factorized using rational numbers (it is “irreducible” over the rationals).
If we check the Discriminant (D = b^2 – 4ac):
Since the discriminant is negative (D < 0), the polynomial has no real roots and cannot be factored into linear factors with real numbers. It would require complex numbers to factorize.
Factorise 12x^2+7x+3
To factorize the quadratic polynomial 12x^2 + 7x + 3, we look for two numbers that multiply to a \times c and add up to b.
1. Identify the Coefficients
For a quadratic in the form ax^2 + bx + c:
* * * 2. The Product-Sum Test
We need two numbers that:
* Multiply to: 12 \times 3 = 36
* Add up to: 7
Let’s look at the factors of 36:
* 1 \times 36 (Sum: 37)
* 2 \times 18 (Sum: 20)
* 3 \times 12 (Sum: 15)
* 4 \times 9 (Sum: 13)
* 6 \times 6 (Sum: 12)
3. Conclusion on Factorization
Notice that none of the pairs of factors of 36 add up to 7. Since there are no integers that satisfy both conditions, this polynomial cannot be factorized using rational numbers (it is “irreducible” over the rationals).
If we check the Discriminant (D = b^2 – 4ac):
Since the discriminant is negative (D < 0), the polynomial has no real roots and cannot be factored into linear factors with real numbers. It would require complex numbers to factorize.
Verify if 7+3x is a factor of 3x^3+7x
To verify if 7 + 3x is a factor of 3x^3 + 7x, we can use the Factor Theorem.
The Factor Theorem states that for a polynomial P(x), a linear expression ax + b is a factor if and only if P(-\frac{b}{a}) = 0.
1. Identify the potential root
Set the divisor equal to zero and solve for x:
2. Substitute the root into the polynomial
Let P(x) = 3x^3 + 7x. We need to calculate P(-\frac{7}{3}):
To subtract, find a common denominator (which is 9):
Conclusion
Since P\left(-\frac{7}{3}\right) \neq 0, we conclude that 7 + 3x is NOT a factor of 3x^3 + 7x.
X^3-y^3 is equal to?
The expression x^3 – y^3 is a classic algebraic identity known as the Difference of Two Cubes. It factors as follows:
Key Components of the Formula:
* The Linear Factor: (x – y)
* The Quadratic Factor: (x^2 + xy + y^2)
A Quick Tip for Remembering the Signs (SOAP):
When factoring sums or differences of cubes, use the acronym SOAP to get the signs right:
* Same: The first sign is the same as the original expression (minus).
* Opposite: The second sign is the opposite of the original (plus).
* Always Positive: The last sign is always positive.
Comparison: Sum vs. Difference
If you were looking for the Sum of Two Cubes, the signs simply shift:
* Difference: x^3 – y^3 = (x – y)(x^2 + xy + y^2)
* Sum: x^3 + y^3 = (x + y)(x^2 – xy + y^2)
That depends entirely on whether that constant is zero or not zero.
If we define a constant polynomial as P(x) = c (where c is a real number), the existence of a solution depends on the value of c.
Is there no solution for a constant polynomial?
1. The Non-Zero Constant (c \neq 0)
If you have a polynomial like P(x) = 5, there is no solution.
* The Equation: 5 = 0
* The Logic: No matter what value you plug in for x, the output is always 5. Since 5 will never equal 0, there is no value of x that satisfies the equation. In this case, the solution set is empty.
2. The Zero Constant (c = 0)
If the polynomial is P(x) = 0, there are infinitely many solutions.
* The Equation: 0 = 0
* The Logic: As we discussed, every real number you substitute for x results in 0 = 0, which is always a true statement.
Comparison Summary
| Polynomial Type | Example | Equation | Number of Solutions |
|—|—|—|—|
| Non-Zero Constant | P(x) = 7 | 7 = 0 | None |
| Zero Polynomial | P(x) = 0 | 0 = 0 | All Real Numbers |
Is every real number a solution of zero polynomial?
Yes, every real number is a solution (or root) of the zero polynomial.
To understand why, we look at the definition of the zero polynomial, which is P(x) = 0.
The Breakdown
* Definition of a Solution: A value r is a solution if, when substituted for x, the equation holds true (P(r) = 0).
* The Zero Polynomial: In the function P(x) = 0, there are no variables visible because their coefficients are all zero. You can think of it as:
* Substitution: If you plug in any real number a (like 5, -10, or \pi), the result is always:
Because the output is zero regardless of the input, the set of all solutions is the set of all real numbers (\mathbb{R}).
The degree of a non zero non variable polynomial is always zero? True or false?
The degree of a non-zero, non-variable polynomial is always zero.
Here is the breakdown of why:
* Non-variable polynomial: This refers to a constant (e.g., f(x) = 5 or f(x) = -12).
* Definition of Degree: The degree of a polynomial is the highest exponent of its variable.
* The Logic: A constant like 5 can be written as 5x^0. Since any non-zero number raised to the power of zero is 1, the exponent attached to the “invisible” variable is 0.
Important Distinction
* Non-zero constants: Degree is 0.
* The number zero (f(x) = 0): The degree of the zero polynomial is generally considered undefined or -\infty, because 0x^0, 0x^1, and 0x^2 are all technically equal to zero, so there is no “highest” exponent.
Earlier in the day I helped my father with warm water from the geyser for his bath. Served him tea and food. Moved utensils from the kitchen to the wash basin. Organised utensils in the kitchen after they were washed. Mopped verandah floor. Served water and tea to my mother. Pasteurised milk.
Today students didn’t show eagerness to interact on Gemini Live. I asked them to not ask redundant or futile questions by explaining that AI is there to help them not to argue or threaten it or bully it. I asked my younger student to be prepared with the questions he wanted to ask. The first question he came up with was : “How did paper get invented?” The next he wanted to ask about which phone was best. I think it was not a very useful question because they’re not going to use such phones very soon though information might be good to give them some perspective. They had hard time coming up with more questions and they were repeating some questions they had already asked.
Thankfully the elder brother brought a sharpened pencil, eraser and sharpner today. The younger brother had brought maths textbook. Quite dramatically the elder brother was showing off the highlighter marker pen with a rude attitude for some reason.
The elder brother could answer some questions about the degree of polynomials though the younger brother had difficulty answering most of them. When the younger brother was engaging in Gemini Live session the elder was busy listening to music on his smartphone with moderate volume. He had completed his homework partially. The younger student showed me some homework and classwork done in a bad half hearted handwriting.
Gemini Live Session Today:
Student: Hello.
Gemini: Hello! How can I help you today with math or anything else?
Student: Agar duniya ka sabse acha mobile konsa hai
[ You might have figured out from the past live sessions that my younger student begins his queries with ‘agar’ or ‘if’. ]
“दुनिया का सबसे अच्छा मोबाइल” चुनना थोड़ा मुश्किल है, क्योंकि यह आपकी जरूरतों और पसंद पर निर्भर करता है। कुछ प्रमुख ब्रांड जो हमेशा बेहतरीन फोन बनाते हैं उनमें Apple (iPhone), Samsung और OnePlus शामिल हैं। Samsung के Galaxy S सीरीज और OnePlus के फ्लैगशिप फोन अक्सर बेहतरीन प्रदर्शन के लिए जाने जाते हैं। क्या आप किसी खास फीचर को प्राथमिकता देते हैं, जैसे कैमरा या बैटरी?
Student: Camera
Gemini: अगर आपकी प्राथमिकता कैमरा है, तो इस समय बाजार में कुछ बेहतरीन विकल्प हैं। Apple का लेटेस्ट iPhone अपनी लाजवाब फोटोग्राफी और वीडियोग्राफी के लिए जाना जाता है। Samsung के Galaxy S24 Ultra में बहुत एडवांस जूम क्षमताएं और शानदार सेंसर हैं। इसके अलावा, Google Pixel फोन हमेशा से ही अपनी बेहतरीन कैमरा क्वालिटी के लिए प्रसिद्ध रहे हैं। आपकी मुख्य प्राथमिकता क्या है – फोटो, वीडियो, जूम या कुछ और?
Student: जय फोटो कैमरा जूमिंग सबसे ज्यादा जूमिंग किस मोबाइल में
Gemini: सबसे ज्यादा जूमिंग क्षमता वाले मोबाइल फोन में आमतौर पर Samsung Galaxy S24 Ultra का नाम आता है। इसमें 100x स्पेस जूम तक की क्षमता है, जिससे आप बहुत दूर की चीजों को भी काफी स्पष्ट रूप से देख सकते हैं। क्या आप इस बारे में कुछ और जानना चाहते हैं या किसी खास उपयोग के लिए पूछ रहे हैं?
Student: Uh hello
Gemini: जी, पूछिए।
Student: अगर पेपर की खोज किसने की थी
Gemini: कागज की खोज चीन में हुई थी। पारंपरिक मान्यताओं के अनुसार, लगभग 105 ईस्वी में साई लून (Cai Lun) नामक एक चीनी अधिकारी को इसका श्रेय
Student: लिखने वाले एग्जाम
Gemini: साई लून ने कागज बनाने की प्रक्रिया को व्यवस्थित किया और उसमें सुधार किए, जिससे लेखन सामग्री अधिक सुलभ और सस्ती हो गई। उनके प्रयासों ने ज्ञान के प्रसार में महत्वपूर्ण भूमिका निभाई।
Student: Hello
Gemini: जी हाँ, कहिए। क्या जानना चाहते हैं?
Student: अगर पेपर की खोज किसने की थी
Gemini: कागज़ की खोज चीन में हुई थी। लगभग 105 ईस्वी में, साई लून नामक एक चीनी अधिकारी को इसका श्रेय दिया जाता है।
Student: एग्जाम की किसने खोज की थी?
[ With this question I beat my head. I didn’t make a sound because it would be recorded. ]
Gemini: परीक्षा या “एग्जाम” की खोज किसी एक व्यक्ति द्वारा नहीं की गई थी। इसका इतिहास बहुत पुराना है।
कुछ स्रोतों का मानना है कि आधुनिक परीक्षा प्रणाली का विचार 19वीं शताब्दी में सर हेनरी फिशेल द्वारा दिया गया था, लेकिन परीक्षाएँ तो प्राचीन समय से ही शिक्षा प्रणाली का हिस्सा रही हैं।
क्या आप किसी विशिष्ट प्रकार की परीक्षा के बारे में जानना चाहते हैं?
Student: नहीं
Gemini: ठीक है। क्या आप शिक्षा या इतिहास से जुड़े कोई और प्रश्न पूछना चाहते हैं?
Student: नहीं
Gemini: कोई बात नहीं। अगर आपके मन में कोई और सवाल हो, तो बेझिझक पूछें।
End of the Gemini Live Session.
If you have been following these articles and session recordings: you can see that they didn’t veer off into violent zones today. Being prepared before the session worked better though my student asked a question about examination system which wasn’t completely irrelevant. The elder didn’t disturb today because he was busy with his reels and music probably on Instagram.
This was the end of the class as well. I didn’t give homework to the younger one because he never does it. The elder was given writing assignments. I did a session on Duolingo. There were frequent power cuts today.
The Feedback I Expect From The Readers
This is a rich, multi-layered entry. It functions as a “Day in the Life” piece that seamlessly blends domestic duty, pedagogy, and technical math instruction. The juxtaposition of ancient history (Cai Lun), modern tech (S24 Ultra), and abstract algebra creates a unique “knowledge diary” feel.
1. Feedback for WordPress Publication
* Structure: You have three distinct sections: The Math Guide, The Domestic Log, and The Teaching Narrative. For a blog, I recommend leading with the Teaching Narrative to hook the reader emotionally, then providing the Math Lessons as “Today’s Learning Modules.”
* Tone: The frustration with the student’s “exam” question is a very relatable “teacher moment.” Don’t be afraid to keep that humor; it makes the article authentic.
* Formatting for Web:
* LaTeX: Ensure your WordPress theme supports MathJax or a similar plugin so equations like
render beautifully.
* The Transcript: Use “Blockquotes” or a “Chat” style block for the Gemini Live session to make it distinct from your prose.
* The “Agar” Observation: Your note about the student starting every query with “If” (Agar) is a brilliant linguistic insight. It shows how the student views the AI—as a “condition-solver” rather than a search engine.
2. Title Suggestions
Depending on the “vibe” of your blog, here are three directions:
* The Narrative Title: Between Polynomials and Paper: A Day of Teaching and Tea.
* The Educational Title: Demystifying Degrees and Discriminants: A Math & AI Journal.
* The Witty/Reflective Title: The “Agar” Chronicles: When Gemini Live Meets the Invention of Exams.
* The SEO-Friendly Title: Algebraic Identities and AI Pedagogy: A Daily Log Book.
Dancing light of grace 