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ما الذي سيتعلمه الطلاب

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ما الذي سيتعلمه الطلاب

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في نهاية الدرس، سيتمكن الطلاب من التعرف على أنواع وطرق مختلفة لتحليل إلى عوامل باستخدام الأساليب الرياضية.

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لماذا يُعتبر تحليل الى عوامل مهمًا في الحسابات الرياضية؟

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Why Do We Break Polynomials Into Factors? | Outlier.org

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التعليم المباشر

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التعليم المباشر

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شرح طرق تحليل الاعداد الى عوامل بالتفصيل باستخدام الأمثلة التطبيقية. استخدام البادلت لعرض ومناقشة الأمثلة.

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Factors

Numbers have factors:

And expressions (like x²+4x+3) also have factors:

Factoring

Factoring (called " Factorising" in the UK) is the process of finding the factors:

Factoring: Finding what to multiply together to get an expression.

It is like "splitting" an expression into a multiplication of simpler expressions.

Example: factor 2y+6

Both 2y and 6 have a common factor of 2:

So we can factor the whole expression into:

2y+6 = 2(y+3)

So 2y+6 has been "factored into" 2 and y+3

Factoring is also the opposite of Expanding:

Common Factor

In the previous example we saw that 2y and 6 had a common factor of 2

But to do the job properly we need the highest common factor, including any variables

Firstly, 3 and 12 have a common factor of 3.

So we could have:

But we can do better!

Together that makes 3y:

So we can factor the whole expression into:

3y ²+12y = 3y(y+4)

Check: 3y(y+4) = 3y × y + 3y × 4 = 3y²+12y

More Complicated Factoring

Factoring Can Be Hard !

The examples have been simple so far, but factoring can be very tricky.

Because we have to figure what got multiplied to produce the expression we are given!

It is like trying to find which ingredients
went into a cake to make it so delicious.
It can be hard to figure out!

Experience Helps

With more experience factoring becomes easier.

Example: Factor 4x² − 9

Hmmm... there don't seem to be any common factors.

But knowing the Special Binomial Products gives us a clue called the "difference of squares":

So we have:

And that can be produced by the difference of squares formula:

So let us try doing that:

Yes!

Answer: 4x ² − 9 = (2x+3)(2x−3)

How can you learn to do that? By getting lots of practice, and knowing "Identities"!

Remember these Identities

Here is a list of common "Identities" (including the "difference of squares" used above).

It is worth remembering these, as they can make factoring easier.

=

(a+b)(a−b)

=

(a+b) (a+b)

=

(a−b) (a−b)

=

=

a ³+3a ²b+3ab ²+b ³

=

(a+b) ³

a ³−3a ²b+3ab ²−b ³

=

(a−b) ³

There are many more like those, but those are the most useful ones.

Advice

The factored form is usually best.

When trying to factor, follow these steps:

There are also Computer Algebra Systems (called "CAS") such as Axiom, Derive, Macsyma, Maple, Mathematica, MuPAD, Reduce and others that can do factoring.

More Examples

Experience does help, so here are more examples to help you on the way:

Example: w⁴ − 16

An exponent of 4? Maybe we could try an exponent of 2:

Yes, it is the difference of squares

And "(w ² − 4)" is another difference of squares

That is as far as I can go (unless I use imaginary numbers)

Remove common factor "3u":

Then a difference of cubes:

That is as far as I can go.

Try factoring the first two and second two separately:

z ²(z−1) − 9(z−1)

Wow, (z-1) is on both, so let us use that:

And is a difference of squares

(z−3)(z+3)(z−1)

That is as far as I can go.

Now get some more experience:

338, 339, 2047, 2048, 2049, 178, 2050, 3177, 3178, 3179

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Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

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التطبيق الموجه

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التطبيق الموجه

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منح الطلاب ورقة عمل تحتوي على مسائل مختلفة لتحليل الأعداد. يتم تقسيم الطلاب إلى مجموعات صغيرة لمناقشة الحلول المشتركة.

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الممارسة المستقلة

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الممارسة المستقلة

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توجيه الطلاب لحل مسائل فردية على تحليل الأعداد إلى عوامل باستخدام تطبيقات تفاعلية.

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Dividing Numbers into Two Factors and Building Arrays to Represent Each Factorization

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تقديم أوراق عمل إضافية للتلاميذ الذين يحتاجون إلى تدريب أكثر أو الذين يريدون تحديات إضافية في تحليل الأعداد المعقدة.

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التقييم التكويني

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التقييم التكويني

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القيام بملاحظة أداء الطلاب أثناء حلهم للمسائل ومناقشتهم للحلول مع زملائهم.

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التقييم الختامي

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التقييم الختامي

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إجراء اختبار قصير للتأكد من فهم الطلاب لمفهوم تحليل الى عوامل باستخدام أوراق عمل معدة مسبقاً.

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Factorization Of Algebraic Expressions Quiz Questions & Answers

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Do you know how to factorize algebraic expressions? Take this Factorization of Algebraic Expressions Quiz to see how well you understand the concept. Factorization involves reducing equations into a simpler form, making it a crucial skill in algebra. This quiz features a variety of questions and answers to test your knowledge and help you practice. Whether you are a student looking to improve your skills or a math enthusiast wanting to challenge yourself, this quiz is perfect for you. You will encounter different types of algebraic expressions and learn the techniques to factorize them efficiently. By taking this quiz, you can identify areas where you need more practice and reinforce your understanding of the concepts. All the best! Go for it and ace it! After completing the quiz, don’t forget to share it with friends, classmates, or anyone interested in mastering algebra. Practicing together can be fun and beneficial for everyone. Start the Factorization of Algebraic Expressions Quiz now and take a step closer to algebra mastery!

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مناقشة جماعية

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مناقشة جماعية

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إجراء مناقشة صفية لتعزيز الفهم واستعراض بعض الأسئلة المفتوحة حول الدرس.

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تذكرة الخروج

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تذكرة الخروج

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طلب من الطلاب كتابة ما تعلموه في الدرس اليومي مع توضيح أي نقاط غموض مازالت لديهم.

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التدريب الإضافي

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التدريب الإضافي

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إنهاء أوراق العمل المتبقية التي لم تُحل في الفصل لتقوية المفاهيم المكتسبة.

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مشروع بحثي صغير

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مشروع بحثي صغير

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تكليف الطلاب بالبحث عن استخدامات عملية لتحليل الأعداد في الحياة اليومية وتقديمها باستخدام وسيلة مثل البادلت.

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What are some real life applications of factorization?

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Answer (1 of 7): Factoring is often a key skill for solving problems in which you need to find a value for x. What can x equal in real life? Well, about anything. Being able to solve for x is the foundation of algebra, which itself is the foundation for doing trigonometry and calculus and higher…