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Algebra - Algebraic Manipulation and Factorization

Grade 8Cambridge (IGCSE)

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Expanding single and double brackets using the distributive law.

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Factorization by taking out the Highest Common Factor (HCF).

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Factorization by grouping terms (usually for expressions with four terms).

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Factorizing quadratic expressions of the form x2+bx+cx^2 + bx + c.

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Recognizing and factorizing the Difference of Two Squares.

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Simplifying algebraic fractions by canceling common factors from the numerator and denominator.

📐Formulae

a(b+c)=ab+aca(b + c) = ab + ac

(x+a)(x+b)=x2+(a+b)x+ab(x + a)(x + b) = x^2 + (a + b)x + ab

a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)

(a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2

(a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2

💡Examples

Problem 1:

Expand and simplify: (2x−3)(x+5)(2x - 3)(x + 5)

Solution:

2x2+7x−152x^2 + 7x - 15

Explanation:

Use the FOIL method (First, Outer, Inner, Last): 2x(x)+2x(5)−3(x)−3(5)=2x2+10x−3x−152x(x) + 2x(5) - 3(x) - 3(5) = 2x^2 + 10x - 3x - 15. Combine the like terms 10x10x and −3x-3x to get 7x7x.

Problem 2:

Factorize completely: 12x2y−18xy212x^2y - 18xy^2

Solution:

6xy(2x−3y)6xy(2x - 3y)

Explanation:

Identify the Highest Common Factor (HCF) of 1212 and 1818, which is 66. For the variables, the HCF of x2x^2 and xx is xx, and the HCF of yy and y2y^2 is yy. Divide each term by 6xy6xy.

Problem 3:

Factorize: x2−49x^2 - 49

Solution:

(x−7)(x+7)(x - 7)(x + 7)

Explanation:

This is a 'Difference of Two Squares' because both x2x^2 and 4949 (727^2) are perfect squares separated by a minus sign. Apply the formula a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b).

Problem 4:

Factorize the quadratic expression: x2−5x+6x^2 - 5x + 6

Solution:

(x−2)(x−3)(x - 2)(x - 3)

Explanation:

Find two numbers that multiply to +6+6 and add to −5-5. Those numbers are −2-2 and −3-3. Therefore, the factors are (x−2)(x - 2) and (x−3)(x - 3).

Problem 5:

Simplify the algebraic fraction: x2−92x+6\frac{x^2 - 9}{2x + 6}

Solution:

x−32\frac{x - 3}{2}

Explanation:

First, factorize the numerator using the difference of two squares: (x−3)(x+3)(x-3)(x+3). Then, factorize the denominator by taking out the common factor 22: 2(x+3)2(x+3). Cancel the common factor (x+3)(x+3) from both the top and bottom.