Calcul littéral

1. Numerical expression and algebraic(literal) expression :

Définition :

▶ A Numerical expression is an expression containing just numbers, which can be calculated 

Exemple :

➤ $-2 \times 5+(5-8)=-10+(-3)=-10-3=-13$

Définition :

▶ An Algebraic(literal) expression is any expression that contains numbers and letters.
▶ Letters symbolize numbers whose value is unknown.

Exemple :

➤ We consider the following algebraic(literal) expression $A=5 x+3$, Calculate the expression A for the values $x=2$ and $x=\frac{-1}{3}$
➤ We Substitute 2 for $\mathbf{X}$ and we have : $\begin{aligned} & A=5 \times 2+3 \\ & A=13\end{aligned}$
➤ We Substitute $\left(\frac{-1}{3}\right)$ for $\mathbf{X}$ and we have : $\begin{aligned} & A=5 \times\left(\frac{-1}{3}\right)+3 \\ & A=\frac{4}{3}\end{aligned}$

Remark :

▸ 4 x means $4 \times \mathrm{x}$
– When the same letter is used several times in an algebraic expression, it always refers to the same number.

2. Literal calculation (arithmetic) :

Définition :

▶ The Literal calculation(arithmetic) is every calculation that uses letters and numbers in the
same expression.

Exemple :

– Calculate the value of the expression : $A=3 \times x+5$ for a given number $x=12$
$\begin{aligned} & A=3 \times 12+5 \\ & A=36+5 \\ & A=41\end{aligned}$

3. Simplifying a literal expression

Définition :

▶ Simplifying a Literal expression is to regroup the same terms with each other (Same letters and
same exponents)

Exemple :

$\begin{aligned} & A=\frac{1}{2} x^2+\frac{7}{5} a+\frac{2}{3} x^2-\frac{2}{5} a \\ & =\frac{1}{2} x^2+\frac{2}{3} x^2+\frac{7}{5} a-\frac{2}{5} a \\ & =\frac{3}{6} x^2+\frac{4}{6} x^2+\frac{5}{5} a \\ & =\frac{7}{6} x^2+a\end{aligned}$

$\begin{aligned} B & =2 a+3 b^2-16 a-7 b+2 b^2 \\ & =2 a-16 a+3 b^2+2 b^2-7 b \\ & =-14 a+5 b^2-7 b\end{aligned}$

1. Definition of expansion(developing) :

Définition :

► Expansion or developing is to transform a product into a sum.
► To do this, we use the distributivity of multiplication to addition.

2. The distributive law (Distributivity of addition over multiplication) :

Rule :

$\mathbf{a , b}$ and $\mathbf{k}$ are rational numbers :
– We have : $\begin{gathered}K(a+b)=K a+K b \\ K(a-b)=K a-K b\end{gathered}$

Exemple :

– Expand the following algebraic expressions :

$$
\begin{gathered}
5(a-b+c-3)=5 \times a-5 \times b+5 \times c-5 \times 3 \\
5 a-5 b+5 c-15 \\
a(x-y)=a \times x-a \times y \\
a x-a y
\end{gathered}
$$

Remark :

▸ We say that the addition is distributive over multiplication
▸ $\mathbf{a , b , c}$ and $\mathbf{k}$ are rational numbers:
▸ We have : $\mathrm{k} \times(\mathrm{a}-\mathrm{b}+\mathrm{c})=\mathrm{k} \times \mathrm{a}-\mathrm{k} \times \mathrm{b}+\mathrm{k} \times \mathrm{c}$

3. The double expansion(developing) or the product of two sums:

Rule :

$\mathbf{a}, \mathbf{b}, \mathbf{c}$ and $\mathbf{d}$ are rational numbers:
– We have :

$$
\begin{aligned}
& \quad A=(a+b)(c+d) \\
& a(c+d)+b(c+d) \\
& a c+a d+b c+b d
\end{aligned}
$$

Exemple :

– $\mathbf{a}$ is a rational number, expand and simplify the expression : $\begin{aligned} & R=(a+1)(3+2 a) \\ & (3+2 a)+1(3+2 a) \\ & a+2 a^2+3+2 a \\ & a^2+5 a+3\end{aligned}$

Rule :

– To multiply a sum by a sum, multiply each term of the first sum by each term of the second sum.
– $\mathbf{a}, \mathbf{b}, \mathbf{c}$ and $\mathbf{d}$ are rational numbers :
➤ $(\mathrm{a}+\mathrm{b})(\mathrm{c}-\mathrm{d})=\mathrm{ac}-\mathrm{ad}+\mathrm{bc}-\mathrm{bd}$
➤ $(\mathrm{a}-\mathrm{b})(\mathrm{c}-\mathrm{d})=\mathrm{ac}-\mathrm{ad}-\mathrm{bc}+\mathrm{bd}$
➤ $(\mathrm{a}-\mathrm{b})(\mathrm{c}+\mathrm{d})=\mathrm{ac}+\mathrm{ad}-\mathrm{bc}-\mathrm{bd}$

Définition :

► Factorizing an algebraic expression means transforming it from a sum to a product.

$$
\begin{aligned}
& K a+K b=K(a+b) \\
& K a-K b=K(a-b)
\end{aligned}
$$

Exemple :

– $\mathbf{a}$ and $\mathbf{b}$ are rational numbers:
– We consider the following expression:
$$
\begin{aligned}
A & =5 a+10 b-20 \\
& =5 \times(a+2 b-4)
\end{aligned}
$$
– We say : We have factorized the algebraic expression $\mathbf{A}$ and the number 5 is : the common factor.

Remark :

1. $1^{\text {st }}$ Remarkable identity (special binomial product)

Rule :

$\mathbf{a , b}$ are rational numbers :
– We have :
$(a+b)^2=(a+b)(a+b)$
$\begin{aligned} & a(a+b)+b(a+b) \\ & a^2+a b+b a+b^2 \\ & a^2+2 a b+b^2\end{aligned}$

Exemple :

– $\mathbf{X}$ is a rational number Expand the following expression :
$\begin{aligned} & L=(x+3)^2 \\ & (x)^2+2 \times(x) \times(3)+(3)^2 \\ & x^2+6 x+9\end{aligned}$
$K=\left(x+\frac{1}{3}\right)^2$
$\begin{array}{r}(x)^2+2 \times(x) \times\left(\frac{1}{3}\right)+\left(\frac{1}{3}\right)^2 \\ x^2+\frac{2}{3} x+\frac{1}{9}\end{array}$
– $\mathbf{X}$ is a rational number Factorize the following expression :
$A=49 x^2+28 x+4 \quad B=\frac{1}{25} x^2+\frac{2}{5} x+1$
$\begin{aligned} & (7 x)^2+2 \times(7 x) \times(2)+(2)^2\left(\frac{1}{5} x\right)^2+2 \times\left(\frac{1}{5} x\right) \times(1)+(1)^2 \\ & (7 x+2)^2\left(\frac{1}{5} x+1\right)^2\end{aligned}$

2. $2^{\text {nd }}$ Remarkable identity (special binomial product)

Rule :

$\mathbf{a}, \mathbf{b}$ are rational numbers :
– We have :

$\begin{aligned} & (a-b)^2=(a-b)(a-b) \\ & a(a-b)-b(a-b) \\ & a^2-a b-b a+b^2 \\ & a^2-a b-a b+b^2 \\ & a^2-2 a b+b^2\end{aligned}$

Exemple :

– X is a rational number Expand the following expression :
$\begin{aligned} & B=(2 x-1)^2 \\ & (2 x)^2-2 \times(2 x) \times(1)+ \\ & 4 x^2-4 x+1\end{aligned}$
$D=\left(3 x-\frac{1}{2}\right)^2$
$(3 x)^2-2 \times(3 x) \times\left(\frac{1}{2}\right)+$
$9 x^2-3 x+\frac{1}{4}$
– X is a rational number Factorize the following expression :
$\begin{aligned} & C=x^2-8 x+16 \\ & (x)^2-2 \times(4) \times(x)+(4)^2\end{aligned}$
$E=\frac{4}{49} x^2-\frac{12}{7} x+9$
$\left(\frac{2}{7} x\right)^2-2 \times\left(\frac{2}{7} x\right) \times(3)+(3)^2$

3. $\mathbf{3}^{\mathrm{rd}}$ Remarkable identity (difference between two squares)s

Rule :

$\mathbf{a}, \mathbf{b}$ are rational numbers :
– We have :
$(a-b)(a+b)=a(a+b)-b(a+b)$
$a^2+a b-b a-b^2$
$a^2+a b-a b-b^2$
$a^2-b^2$

Exemple :

➧ X is a rational number Expand the following expression :
$\begin{aligned} & D=(3 x+5)(3 x-5) \\ & 3 x \times(3 x-5)+5 \times(3 x-5) \\ & \frac{16}{9} x^2-49 \\ & 9 x^2-15 x+15 x-25 \\ & 9 x^2-25 \\ & P=\left(\frac{4}{3} x+7\right)\left(\frac{4}{3} x-7\right) \\ & \quad\left(\frac{4}{3} x\right)^2-(7)^2\end{aligned}$
– $\mathbf{X}$ is a rational number Factorize the following expression :
$\begin{gathered}E=x^2-25 \\ (x)^2-(5)^2 \\ (x-5)(x+5) \\ W=4 x^2-\frac{1}{9} \\ (2 x)^2-\left(\frac{1}{3}\right)^2 \\ \left(2 x-\frac{1}{3}\right)\left(2 x+\frac{1}{3}\right)\end{gathered}$

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