Calcul littéral
I-Numerical expression and algebraic(literal) expression
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 :
$\begin{aligned} & A=\frac{1}{2} x^2+\frac{7}{5} a+\frac{2}{3} x^2-\frac{2}{5} a \\ & =\frac{1}{2} x^2+\frac{2}{3} x^2+\frac{7}{5} a-\frac{2}{5} a \\ & =\frac{3}{6} x^2+\frac{4}{6} x^2+\frac{5}{5} a \\ & =\frac{7}{6} x^2+a\end{aligned}$
$\begin{aligned} B & =2 a+3 b^2-16 a-7 b+2 b^2 \\ & =2 a-16 a+3 b^2+2 b^2-7 b \\ & =-14 a+5 b^2-7 b\end{aligned}$
II- Expansion(developing) of a literal expression and the distributive law
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 :
– 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 :
➤ $(\mathrm{a}+\mathrm{b})(\mathrm{c}-\mathrm{d})=\mathrm{ac}-\mathrm{ad}+\mathrm{bc}-\mathrm{bd}$
➤ $(\mathrm{a}-\mathrm{b})(\mathrm{c}-\mathrm{d})=\mathrm{ac}-\mathrm{ad}-\mathrm{bc}+\mathrm{bd}$
➤ $(\mathrm{a}-\mathrm{b})(\mathrm{c}+\mathrm{d})=\mathrm{ac}+\mathrm{ad}-\mathrm{bc}-\mathrm{bd}$
III- Factorization of a literal expression
Définition :
Exemple :