Page 2: Advanced Algebraic Operations and Notable Identities

This page delves deeper into Développer et réduire une expression en ligne, focusing on more complex expansions and the third remarkable identity.

The page starts with a detailed example of expanding and reducing: 2x + 3$$x + 5 = 2x² + 10x + 3x + 15 = 2x² + 13x + 15 7x + 6$$2x - 1 = 14x² - 7x + 12x - 6 = 14x² + 5x - 6

Example: When expanding 2x + 3$$x + 5, we multiply each term of the first bracket by each term of the second: 2x × x, 2x × 5, 3 × x, and 3 × 5.

The page then introduces the third remarkable identity: a + b$$a - b = a² - b²

Highlight: The third remarkable identity is crucial for simplifying certain types of expressions quickly.

Examples of applying this identity are provided: 5x + 6$$5x - 6 = (5x)² - 6² = 25x² - 36 9x + 3$$9x - 3 = (9x)² - 3² = 81x² - 9

Vocabulary: A remarkable identity is a standard algebraic formula that can be applied to simplify expressions quickly.

The page concludes with an example of factoring using the third remarkable identity: 36x² - 16 = (6x)² - 4² = 6x + 4$$6x - 4

Definition: Factoring using the third remarkable identity involves recognizing the pattern a² - b² and rewriting it as a + b$$a - b.

This comprehensive guide provides students with a solid foundation in algebraic operations, preparing them for more advanced mathematical concepts.

Page 1: Basic Algebraic Operations

This page introduces fundamental algebraic operations essential for Développer et réduire 3ème level mathematics. It covers reducing expressions, removing parentheses, expanding, and factoring.

Definition: Reducing an algebraic expression means simplifying it by combining like terms.

The page begins with examples of reducing expressions: $5x² - 6x + 3$ + $- 7x - 2x²$ = 3x² - 13x - 7 $8x² - 30x²$ + 6ax - 40a² = -22x² + 6ax - 40a²

Example: When reducing $5x² - 6x + 3$ + $- 7x - 2x²$, we combine like terms: 5x² and -2x² become 3x², -6x and -7x become -13x, and 3 remains as is.

Next, it demonstrates reducing expressions while removing parentheses: -$2x + 3$ + $7x - 9$ = -2x - 3 + 7x - 9 = 5x - 12 5x - $-4x - 7$ - $9x + 6$ = 5x + 4x + 7 - 9x - 6 = 1 $-5x$ - $-7 + 3x$ = -5x + 7 + 3x = -2x + 7

Highlight: When removing parentheses, pay attention to the signs. A minus sign before parentheses changes all signs inside.

The page then moves on to expanding expressions: 5x$4x + 1$ = 20x² + 5x $-5x$ × $-7 + 3x$ = 35x - 15x² = -15x² + 35x

Vocabulary: Expanding an expression means multiplying each term inside the parentheses by the term outside.

Finally, it covers factoring: 3x - 3y = 3$x - y$ 8x² + 5x = x$8x + 5$

Definition: Factoring is the process of finding what you can multiply together to get the expression.