Expansion (Development)

This page focuses on the process of expansion, which is the opposite of factorization. It involves multiplying out factors to obtain a sum of terms.

Definition: Expansion, also known as development, is the process of multiplying out factors in an algebraic expression to obtain a sum of terms.

The page starts by introducing the distributive property, which is fundamental to expansion:

k$a + b$ = ka + kb k$a - b$ = ka - kb

Example: A = 5$2x + 3$ A = 5 × 2x + 5 × 3 A = 10x + 15

The text then moves on to the double distributive property, which is used when multiplying two binomials:

$a + b$$c + d$ = ac + ad + bc + bd

Example: $x + 5$$x - 2$ = x × x + x × (-2) + 5 × x + 5 × (-2) = x² - 2x + 5x - 10 = x² + 3x - 10

The page also revisits notable identities, this time from the perspective of expansion:

Highlight: $a + b$$a - b$ = a² - b²

This identity is demonstrated with examples:

Example: $x + 2$$x - 2$ = x² - 2² = x² - 4 Example: $4 - x$$x + 4$ = 4² - x² = 16 - x²

The page concludes with a more complex example that combines various techniques:

2$x + 3$ + $2x + 3$$2x - 3$ = 2x + 6 + (2x)² - 3² = 2x + 6 + 4x² - 9 = 4x² + 2x - 3

This comprehensive coverage of développement et factorisation exercices corrigés PDF provides students with a solid foundation in these essential algebraic techniques. The material is particularly useful for those studying factorisation et développement 3ème and développement et factorisation Seconde, as it covers topics typically found in these grade levels.

Factorization

Factorization is a fundamental algebraic technique that involves expressing a polynomial as a product of its factors. This page covers various methods of factorization, including common factor extraction and notable identities.

Definition: Factorization is the process of breaking down a polynomial into simpler expressions that, when multiplied together, produce the original polynomial.

The page begins by showing an example of factorization and its reverse process, expansion:

x² - 3x + 2 = $x + 2$$x - 1$

It then introduces the common factor method:

Example: 5x² + 35x = 5x$x + 7$

The text also covers factoring by grouping, illustrated with this example:

$2a - 1$$a + 3$ - $a + 3$$2a - 5$ = $a + 3$$(2a - 1) - (2a - 5)$ = $a + 3$$-2a + 4$

Highlight: One of the most important concepts introduced is the notable identity for the difference of squares:

a² - b² = $a + b$$a - b$

This identity is demonstrated with practical examples:

Example: x² - 25 = x² - 5² = $x + 5$$x - 5$ Example: 9x² - 16 = (3x)² - 4² = $3x + 4$$3x - 4$

These examples show how to apply the difference of squares identity to simplify expressions and solve problems more efficiently.