Calculations on the
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Calculations
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Created it, 05/10/15
Update it, 06/01/10
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FIRST PART : ALGEBRA
We now approach our third lesson of mathematics. It is composed of two great parts.
First is devoted to the algebra. You will know what is an algebraical expression and will learn how to carry out calculations on the students'rag processions and the polynomials. You will see then what is an equation and how to solve it.
In the second part, it will be a question of the logarithms. You will see that by their intermediary complex calculations with the powers and the extractions of root are transformed into simple multiplications or divisions.
In addition, as you are now accustomed to the mathematical developments, we will preferably employ them from now on with the literal explanations increasingly longer and less precise.
Lastly, as we already said and repeated in the first two lessons, we invite you to remake the exercises quoted in examples, as well during your readings as some time afterwards.
1. 1. DEFINITION
One calls algebraical expression, a whole of letters and numbers connected to each other by signs indicating the operations to be carried out.
Example :
Each letter represents a number. If the same letter appears several times in the same expression, it represents the same number there. To obtain the numerical value of the expression, it is enough to replace each letter by the number which it represents.
Example : calculate the numerical value of :
When a = 1 and b = 2
We write :
One needs care and attention. All that is in a bracket, will have to be considered a single number to take as such, after having calculated it.
The rule of the signs, which we saw, applies obviously in these calculations. In addition, the rule concerning the removal of the brackets preceded by the sign about also applies:
Example : if a = 1 and b = 2, one obtains :
The first two expressions which differ only by brackets and signs have, as you see it, of the different numerical values.
1. 3. - VARIOUS FORMS OF ALGEBRAICAL EXPRESSIONS
1. 3. 1. - ALGEBRAICAL EXPRESSION KNOWN AS WHOLE
The expression does not contain letters with the denominator.
Example :
1. 3. 2. - ALGEBRAICAL EXPRESSION KNOWN AS FRACTIONAL
The expression contains letters with the denominator.
Example :
1. 3. 3. - Students'rag processions
The expression does not contain signs of addition or subtraction.
Example :
1. 3. 4. - POLYNOMIALS
The expression contains signs of addition or subtraction between several students'rag processions.
Example :
2. - CALCULATIONS
ON THE STUDENTS'RAG PROCESSIONS2. 1. - WRITING OF A STUDENTS'RAG PROCESSION
Since a students'rag procession is a product of factors, and that one can invert the order of his factors, without changing the result, it is always necessary to arrange oneself to reduce the students'rag processions in a condensed form more easily usable :
Example :
A students'rag procession is composed of two part :
Examples :
3a2b 3 is the coefficient and a2b is the literal part ;
a2b 1 is the coefficient and a2b is the literal part ;
- a2b - 1 is the coefficient and a2b is the literal part.
(For the two last expressions, coefficient 1 is implied).
2. 2. 1. - DEGREE OF A STUDENTS'RAG PROCESSION COMPARED TO A LETTER
Definition : One calls degree of a students'rag procession compared to a letter, the exhibitor of this letter in this students'rag procession.
3a2b is degree 2 for (a) and degree 1 for b.
x3y4 is degree 3 for (x) and of degree 4 for y.
2. 2. 2. - DEGREE OF A STUDENTS'RAG PROCESSION COMPARED TO A WHOLE OF LETTERS
Definition : One calls degree of a students'rag procession compared to the whole of the letters, the sum of the exhibitors of all his letters.
The students'rag procession 2a2bx3y4 is degree 10 (2 + 1 + 3 + 4) for the whole of its letters.
2. 3. - SIMILAR STUDENTS'RAG PROCESSIONS
Definition : Similar students'rag processions are students'rag processions which have even literal part.
Examples :
It results from this immediately that the sum of several similar students'rag processions is a similar students'rag procession whose coefficient is the sum of the coefficients of the students'rag processions:
It is what one invites to reduce the similar students'rag processions.
2. 4. 1. - PRODUCT OF SEVERAL STUDENTS'RAG PROCESSIONS
The product of several students'rag processions is a students'rag procession :
- whose coefficient is the product of the coefficients of the students'rag processions given ;
- the literal part includes/understands the letters contained in the students'rag processions, each one of them being affected of an exhibitor equal to the sum of his exhibitors in the factors.
Example :
2. 4. 2. - QUOTIENT OF STUDENTS'RAG PROCESSIONS
The quotient of a students'rag procession by a students'rag procession is written in the form of a fraction which it is necessary to simplify to the maximum.
examples :
One simplified numerator and denominator by the general term B.
One simplified numerator and denominator by 2ab2c2.
Note: A students'rag procession A is divisible by a number B, when A contains all the letters of B with exhibitors with the least equal.
Examples :
3. - CALCULATIONS ON THE POLYNOMIALS3. 1. - DEFINITIONS
3. 1. 1. - POLYNOMIAL
A polynomial is a sum of several students'rag processions which are the terms of the polynomial.
Examples :
3. 1. 2. - BINOMIAL
One calls binomial, a polynomial which contains only two terms.
Example : 3a + 4b
3. 1. 3. - TRINOMIAL
One calls trinomial, a polynomial which contains only three terms.
Example: 2x2 - 3xy + 4y2
3. 2. - REDUCTION OF POLYNOMIALS
It is always necessary to start by making the polynomial simplest possible.
Example :
While reducing, one finds :
The reduced polynomial will thus be written: 5x3 + x2y + 4xy2 whose form is anyway simpler than that proposed above.
Note : It is the rule to write a polynomial so that the degrees of its terms, compared to one of its letters, are either decreasing, or while increasing.
Examples :
- 8ax3 + 6bx2 + 3cx is ordered compared to the decreasing powers of x.
- 3 + 2xy + (4 / 3) . xy2 - 6y3 is ordered compared to the increasing powers of y.
In addition, one can order a polynomial (or a students'rag procession) according to the alphabetical order of his letters.
Examples :
Instead of : 3ayx + 4yxz - 3bac
One will write : 3axy + 4xyz - 3abc
and better : - 3abc + 3axy + 4xyz
3. 3. - DEGREE OF A POLYNOMIAL
Definition: The degree of a polynomial compared to a letter is the highest exhibitor of this letter in the polynomial.
3. 3. 1. - POLYNOMIAL A ONLY ONE LETTER
Examples :
3. 3. 2. - POLYNOMIAL SEVERAL LETTERS
Examples :
3. 4. - OPERATIONS ON THE POLYNOMIALS
3. 4. 1. - ADDITION OF POLYNOMIALS
Regulate : The sum of several polynomials is obtained by writing the terms of the polynomials the ones following the others and by reducing the similar terms of the polynomial obtained.
Example :
(x4 + 3x2y2 + 3y) + (3x4 + 2x2y3 + 5y) = x4 + 3x2y2 + 3y + 3x4 + 2x2y3 + 5y = x4 + 3x2y2 + 2x2y3 + 8y
3. 4. 2. - SUBTRACTION OF POLYNOMIALS
Regulate: To cut off a polynomial, one adds the terms of this polynomial changed sign.
Examples :
1) (x4 + 3x2y) - (2x4 - 4x2y) = x4 + 3x2y - 2x4 + 4x2y = - x4 + 7x2y
2) (3ab2 + 2a2b) - (2ab2 + 2a2b) = 3ab2 + 2a2b - 2ab2 - 2a2b = ab2
3. 4. 3. - PRODUCT Of a POLYNOMIAL BY a STUDENTS'RAG PROCESSION
Regulate : To multiply a polynomial by a students'rag procession, one multiplies successively each term of the polynomial by the students'rag procession. It is the product of a sum by a number.
Example :
(2x3 - x2 + 2) . (3xy) = 6x4y - 3x3y + 6xy
3. 4. 4. - PRODUCT Of a POLYNOMIAL BY a POLYNOMIAL
A polynomial being the sum of several students'rag processions, one will observe the rule of the multiplication of a sum by a sum.
Regulate: To multiply two polynomials between them, one successively multiplies each term of the one by each term of the other and one algebraically adds the products obtained. Then the similar terms are reduced.
Example :
|
2a2b2 - 3a2b + ab2 |
+ 4a2b - 6a2 + 2ab |
- 2ab2 + 3ab - b2 |
|
Product by ab |
Product by 2a |
Product by - b |
And while reducing : 2a2b2 + a2b - ab2 - 6a2 + 5ab - b2
3. 4. 5. - REMARKABLE PRODUCTS
There are some remarkable products which it is desirable to know by heart.
Square of the sum of two numbers :
Square of the difference in two numbers :
Product of the sum of two numbers by their difference :
Other remarkable products are important :
and
(a + b) (a2 - ab + b2) = a3 + b3
like
(a + b)3 = a3 + 3a2b + 3ab2 + b3
and
Note : The use of the remarkable products often leads to the decomposition of a polynomial in simpler products of factors which are likely later reductions.
Let us take an example, that is to say :
The numerator and the denominator are remarkable products. We can thus write :
and while simplifying by a + b we find :
Expression simpler than that proposed.
3. 4. 6. - DIVISION OF A POLYNOMIAL BY A STUDENTS'RAG PROCESSION
The quotient of the polynomial P = 10x3 - 4x2y + 6xy2 by the students'rag procession 2x is the polynomial which it is necessary to multiply by 2x to obtain the polynomial P. One writes :
A polynomial is thus divisible by a students'rag procession when all the terms of this polynomial are divisible by this students'rag procession.
Regulate: To divide a polynomial by a students'rag procession, one divides all the terms of this polynomial by the students'rag procession.
Application :
1 - Common factorization.
Let us consider the polynomial :
25ax4 + 35ay4 - 55ax2y2
All his terms being divisible by 5a, one can write it in the form :
It is said that the students'rag procession (5a) was put in common factor in the polynomial.
The students'rag procession of higher degree which can be put in factor in a polynomial includes/understands the letters common to all the terms, each letter being affected smaller exhibitor than it has in the polynomial. The coefficient of the students'rag procession put in factor can be arbitrary (example 1), but one generally takes for coefficient highest common factor of the coefficients of the terms (example 2).
Example 1 :
We took (x) as common factor, whereas we could have taken xy.
Example 2 :
2 - Decomposition of a polynomial in a product of factors.
This decomposition is possible :
a - putting a students'rag procession in common factor
b - by grouping the terms of the polynomial so as to be able to then carry out factorizations common.
Example :
Let us put (b) in common factor in the first sum and (y) in common factor in the second:
Let us put (a + x) in factor:
c - by applying the properties of the remarkable products.
Example 1 :
Example 2 :
18abx2 - 12abx + 2ab = 2ab (9x2 - 6x + 1) = 2ab (3x - 1) ; application from (a - b)2
Example 3 :
= (x + y)2 - z2 ; application from (a + b)2
= (x + y + z) (x + y - z) ; application of (a + b) (a - b)
The decomposition of the polynomials in products of factors is often applied to the simplification of the fractions.
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