Expanding Multiple Brackets

When dealing with multiple brackets that need to be multiplied, each term from one bracket must be multiplied by every term in the other bracket. This process is known as Ausmultiplizieren in German.

Example: $4x+3$(2x − 1) = (4x·2x) + $4x· (-1)$ +(3·2x) + (3· (−1)) = 8x² - 4x +6x-3 = 8x²+2x-3

This example demonstrates the step-by-step process of expanding two brackets:

1. Multiply the first terms: 4x · 2x = 8x²
2. Multiply outer terms: 4x · $-1$ = -4x
3. Multiply inner terms: 3 · 2x = 6x
4. Multiply last terms: 3 · $-1$ = -3
5. Combine like terms: 8x² + $-4x + 6x$ - 3 = 8x² + 2x - 3

Highlight: When expanding brackets, be meticulous about keeping track of signs and combining like terms at the end.

The guide provides another complex example to reinforce the concept:

Example: -5y-6$$3x-2y = $-5y· 3x$ + $(-5y)· (-2y)$ + $(-6)· 3x$ + $(-6)·(-2y)$ = -15xy + 10y² - 18x + 12y

This example showcases the importance of carefully applying the rules for sign multiplication when expanding brackets.

Definition: Klammern auflösen (solving brackets) is the process of simplifying algebraic expressions by removing brackets according to specific rules for addition, subtraction, and multiplication.

Practice Exercises and Solutions

The guide concludes with a set of practice exercises to help students apply the rules they've learned. These exercises cover various scenarios of Klammern auflösen, including simple bracket removal, distribution of factors, and expansion of multiple brackets.

Example: Exercise 1: 5x +3 - $7y + 2$ Solution: 5x +3-7y-2 = 5x +1-7y

This example demonstrates the application of the subtraction rule, where signs within the bracket are reversed.

Example: Exercise 2: 6$2x +3$ Solution: 6·2x + 6·3 = 12x +18

This exercise shows the distribution of a factor to terms within a bracket.

Example: Exercise 3: 7x-2$$2-4y Solution: 7x·2+7x· $-4y$ + $-2$·2 + (−2)·$-4y$ = 14x-28xy -4 +8y

This more complex example requires expanding two brackets, carefully considering sign rules.

Highlight: Practice is key to mastering Klammern auflösen. Work through these exercises methodically, paying close attention to signs and the specific rules for each operation.

By providing a mix of examples and exercises, the guide offers students the opportunity to reinforce their understanding of Klammern auflösen and develop confidence in applying these crucial algebraic techniques.

Klammern auflösen: Rules and Examples

Addition and Subtraction with Brackets

When dealing with brackets in addition, the process is straightforward. If a plus sign precedes the bracket, you can simply remove the bracket without changing any signs inside.

Example: 5+7x +$3-4y$ = 5+7x +3-4y

For subtraction, the process requires more attention. When a minus sign precedes the bracket, all signs within the bracket must be reversed when removing it.

Example: 3x+4-$2x -5$ = 3x+4-2x +5

Highlight: Remember the crucial rule: when subtracting a bracketed expression, reverse all signs inside the bracket.

Multiplication with Brackets

When a factor is placed before a bracket, it must be distributed to every term inside the bracket.

Example: 5x+3$2x-4$ = 5x +3·2x +3·$-4$ = 5x+6x-12

Vocabulary: Ausmultiplizieren (expanding brackets) refers to the process of multiplying terms in different brackets.

The guide provides essential rules for various bracket scenarios:

• a$b+c$ = ab + ac
• -a$b+c$ = -ab - ac
• a$b-c$ = ab - ac
• -a$b-c$ = -ab + ac

Highlight: When multiplying brackets, pay close attention to signs. Remember that (-) · (-) = +, (+) · (+) = +, and (-) · (+) = -.