We are now entering our third lesson in mathematics. It is composed of two important parts.
The first is devoted to algebra. You will know what an algebraic expression is and learn how to perform calculations on monomials and polynomials. You will then see what an equation is and how to solve it.
In the second part, we will discuss logarithms. You will see that through them complex calculations with powers and root extractions are transformed into simple multiplications or divisions.
On the other hand, as you are now accustomed to mathematical developments, we will now use them instead of literal explanations that are always longer and less precise.
Finally, as we have already said and repeated in the first two lessons, we invite you to repeat the exercises mentioned in examples, both during your readings and some time later.
1. - ALGERIA EXPRESSIONS
1. 1. DEFINITION
An algebraic expression is a set of letters and numbers linked together by signs indicating the operations to be performed.
Example :
Each letter represents a number. If the same letter appears multiple times in the same expression, it represents the same number. To obtain the numeric value of the expression, simply replace each letter by the number it represents.
Example: calculate the numeric value of :
Whena
= 1 and b = 2
We write :
3 x (1)2 x (2) + 4 x (1) x (2)2 / (1)2 + (2)2 = 6 + 16 / 1
+ 4 = 22 / 5 = 4,4
1. 2. - CALCULATION OF THE DIGITAL VALUE OF AN ALGERIA EXPRESSION
It takes care and attention. Everything in a parenthesis, should be considered a unique number to take as such, after having calculated.
The rule of signs, which we have seen, obviously applies in these calculations. On the other hand, the rule concerning the deletion of parentheses preceded by the + or - sign also applies :
Example:if
a = 1 and b = 2, we obtain :
(a + b) X (a - b) = (1 + 2) X (1 - 2) = 3 X (- 1) = - 3
Since a monomial is a product of factors, and that one can invert the order of its factors, without changing the result, one must always arrange to reduce the monomials in a condensed form more easily usable :
Example :
3 . a . 5 . 2 . b . y2
3 . 5 . 2 . a . b . y2 = 30aby2
A monomial is composed of two parts :
A numerical factor called coefficient;
A product of literal factors called literal 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 last two expressions, the coefficient 1 is implied).
2. 2. - DEGREE OF A WHOLE MONOME
2. 2. 1. - DEGREE OF A MONOME IN RELATION TO A LETTER
Definition :
We call the degree of a monomial in relation to a letter, the exponent of this letter in this monome.
3a2b
is of degree 2 for (a) and of
degree 1 for b.
x3y4is of degree 3 for (x) and of degree 4 for y.
2. 2. 2 . - DEGREE OF A MONOME IN RELATION TO A SET OF LETTERS
Definition : The degree of a monomial in relation to the set of letters is the sum of the exponents of all its letters.
The monomial 2a2bx3y4
is of degree 10 (2 + 1 + 3 + 4)
for all of its letters.
2. 3. - MONOMES SIMILAR
Definition :
Similar monomials are monomials that have even literal part.
Examples :
3a2b ; 4a2b
; - 8a2b are similar monomials
It follows immediately that the sum of several similar monomials is a similar monomial whose coefficient is the sum of the coefficients of the monomials :
3a2b + 4a2b
- 8a2b = (3 + 4 - 8) . a2b = - a2b
This is called reducing similar monomials.
2. 4. - OPERATIONS ON THE MONOMES
2. 4. 1. - PRODUCT OF SEVERAL MONOMES
The product of several monomials is a monomial :
- whose coefficient is the product of the coefficients of the given monomials ;
- whose literal part comprises the letters contained in the monomials, each of them being assigned an exponent equal to the sum of its exponents in the factors.
Example :
3a2b . 4b2c . - 5bd = - 60a2b4cd
Coefficient : (3) . (4) . (- 5) = - 60
Degree for a : 2
Degree for b : 1 + 2 + 1 = 4
Degree for c : 1
Degree for d : 1
For the group : 2 + 4 + 1 + 1 = 8
2. 4. 2. - QUOTIENT OF MONOMES
The quotient of a monomial by a monomial is written in the form of a fraction which must be simplified to the maximum.
examples :
3a2b : 4b2c = 3a2b / 4b2c = 3a2 / 4bc
We have simplified numerator and denominator by the common term b.
4ab2c3
: 2a2b2c2 = 4ab2c3 / 2a2b2c2
= 2 (c / a)
We have simplified numerator and denominator by
2ab2c2.
Remark :
A monomial A is divisible by a number B, when A contains all the letters of B with exponents at least equal.
A polynomial is a sum of several monomials that are the terms of the polynomial.
Examples :
3ac2 + bc3 + 4 ; 2 / 5 . x2 + 3ax2 - 2 / 3 . x
3. 1. 2. - BINOMIAL
A binomial is a polynomial that contains only two terms.
Example : 3a + 4b
3. 1. 3. - TRINOMIAL
A trinomial is a polynomial that contains only three terms.
Example :2x2 - 3xy + 4y2
3. 2. - POLYNOMIAL REDUCTION
Always start with Making the polynomial as simple as possible.
Example :
3x3 + 5x2y
+ 2xy2 + 2x3 - 4x2y + 2xy2
By reducing, we find :
3x3 + 2x3
= 5x3
5x2y - 4x2y = x2y
2xy2 + 2xy2 = 4xy2
The reduced polynomial will therefore write :
5x3 + x2y + 4xy2 dont
whose form is nevertheless simpler than the one proposed above.
Remark : It is the rule to write a polynomial so that the degrees of its terms, with respect to one of its letters, go either decreasing or increasing.
Examples :
- 8ax3 + 6bx2
+ 3cx is ordered relative to the
decreasing powers of x.
- 3 + 2xy + (4 / 3) . xy2
- 6y3 is ordered in relation to the
increasing powers of y.
On the other hand, we can order a polynomial (or a monomial) according to the alphabetical order of its letters.
Examples :
Instead of:
3ayx + 4yxz - 3bac
We will write: 3axy + 4xyz - 3abc
and better:- 3abc + 3axy + 4xyz
3. 3. - DEGREE OF A POLYNOMIAL
Definition :The degree of a polynomial with respect to a letter is the highest exponent of that letter in the polynomial.
3. 3. 1.
- POLYNOMIAL ONLY ONE LETTER
Examples :
2a + 3 is a first degree binomial in a ;
3a2 + 2a - 4 is a trinomial of second degree in a ;
8x2 - 3 is a second degree binomial in x.
3. 3. 2. - POLYNOMIAL WITH MANY LETTERS
Examples :
x4 - 2xy3 est de degré 4 pour x et 3 pour y.
3. 4. - OPERATIONS ON THE POLYNOMIALS
3. 4. 1. - ADDITION OF THE POLYNOMIALS
Rule :The sum of several polynomials is obtained by writing the terms of the polynomials one after the other and reducing the similar terms of the polynomial obtained.
Rule :
To multiply a polynomial by a monomial, we successively multiply each term of the polynomial by the monomial.
It is the product of a sum by a number.
Example :
(2x3-
x2+ 2) . (3xy) = 6x4y - 3x3y
+ 6xy
3. 4. 4.
- PRODUCT OF A POLYNOME BY A POLYNOME
A polynomial being the sum of several monomials, we will apply the rule of the multiplication of a sum by a sum.
Rule :To multiply two polynomials between them, we multiply each term of one successively by each term of the other and we add algebraically the products obtained. Then we reduce the similar terms.
Example :
(2ab - 3a + b) . (ab + 2a - b) =
2a2b2
- 3a2b + ab2
+ 4a2b - 6a2
+ 2ab
- 2ab2 + 3ab
- b2
Produced by ab
Produced by 2a
Produced by - b
And by reducing
: 2a2b2 + a2b - ab2
- 6a2 + 5ab - b2
3. 4. 5. - REMARKABLE PRODUCTS
There are some remarkable products that it is desirable to know by heart.
Square of the sum of two numbers :
(a + b)2 = (a + b) (a + b) = a2 + 2ab + b2
Square of the difference of two numbers :
(a - b)2 = (a - b) (a - b) = a2 - 2ab + b2
Product of the sum of two numbers by their difference :
(a + b) (a - b) = a2 - b2
Other remarkable products are important:
(a - b) (a2 + ab + b2) = a3 - b3
and
(a + b) (a2 - ab + b2) = a3 + b3
as well as
(a + b)3 = a3 + 3a2b + 3ab2 + b3
and
(a - b)3 = a3 - 3a2b + 3ab2 + b3
Remark : The use of remarkable products often leads to the decomposition of a polynomial into products of simpler factors that are susceptible to subsequent reductions.
Let's take an example, that is :
The numerator and the denominator are remarkable products. So we can write :
and simplifying by
a + b we find :
Simpler expression than the one proposed.
3. 4. 6. -DIVISION OF A POLYNOME BY A MONOME
The quotient of the polynomial
P = 10x3 - 4x2y + 6xy2by the monomial 2x
is the polynomial that must be multiplied by 2x
to obtain the polynomial P. We write :
A polynomial is therefore divisible by a monomial when all the terms of this polynomial are divisible by this monomial.
Rule :
To divide a polynomial by a monomial, we divide all the terms of this polynomial by the monomial.
Application :
1 -Common factorization.
Consider the polynomial :
25ax4 + 35ay4 - 55ax2y2
All its terms being divisible by
5a,
we can write it in the form :
5a (5x4 + 7y4 - 11x2y2)
It is said that the monomial (5a)
has been put into common factor in the polynomial.
The higher-order monomial that can be factored into a polynomial includes the letters common to all terms, each letter being assigned the smallest exponent it has in the polynomial. The coefficient of the monomer put in factor can be arbitrary (example 1), but one takes more often for coefficient the greatest common divisor of the coefficients of the terms (example 2).
Example 1:
18x3y - 11x2y2
+ 22xy3 = x (18x2y - 11xy2 + 22y3)
We took (x)
as a common factor, whereas we could have taken xy.
Example 2:
12x2y3
+ 15x3y2 = 3x2y2 (4y + 5x)
2 - Decomposition of a polynomial into a product of factors.
This decomposition is possible :
a - putting a monome in common factor
b - grouping the terms of the polynomial so that it can then perform common factorization.
Example:
ab + bx + ay + xy = (ab + bx) + (ay + xy)
Let (b)
be a common factor in the first sum and (y)
as a common factor in the second :
b (a + x) + y (a + x)
Let's put (a + x)
in factor :
(a + x) (b + y)
c - applying the properties of remarkable products.
Example 1 :
4x2 + 20x + 25 = (2x + 5)2 ; application of (a + b)2
Calculations on the monomials
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