Factorization Techniques
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Factorization Techniques — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionFactoring Difference of SquaresConcept Practice
2 marks~3 minCriterion A
An architect designs a square marble tile with side length a=9a = 9 cm. A square notch of side length b=4b = 4 cm is cut from one corner, creating an L-shaped tile. The area of the L-shaped tile is given by a2−b2a^2 - b^2.
a
Show that a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b), and hence calculate the area of the L-shaped tile. [1]
b
A box holds exactly 10 L-shaped tiles. Justify whether the total tiled area is sufficient to cover a surface of 640 cm2640 \text{ cm}^2. [1]
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Solutions

2QuestionFactoring Trinomials Quadratic FormAssessment Practice
4 marks~6 minCriterion D
A factory produces rectangular metal sheets. The length of each sheet is (x+4)(x + 4) cm and the width is (x+1)(x + 1) cm, where xx is a positive integer. The area AA cm2^2 of each sheet is modelled by A=x2+5x+4A = x^2 + 5x + 4.

Due to a material constraint, each sheet must have an area of exactly 40 cm2^2.
a
Factorise x2+5x+4x^2 + 5x + 4. Hence solve x2+5x+4=40x^2 + 5x + 4 = 40, stating both solutions. [2]
b
Deduce which value of xx is valid in this context and state the corresponding length and width of the sheet. [1]
c
Discuss two limitations of using this algebraic model to represent real-world sheet production. [1]
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Solutions

3QuestionFactoring Difference of SquaresConcept Practice
2 marks~3 minCriterion B
Consider the following factorised expressions:

x2−9=(x−3)(x+3)x^2 - 9 = (x-3)(x+3)
4x2−25=(2x−5)(2x+5)4x^2 - 25 = (2x-5)(2x+5)
9y2−16=(3y−4)(3y+4)9y^2 - 16 = (3y-4)(3y+4)
25m2−49=(5m−7)(5m+7)25m^2 - 49 = (5m-7)(5m+7)
a
Identify the structural pattern linking each expression to its factors. [1]
b
Deduce a general rule for factoring any expression of the form a2−b2a^2 - b^2. [1]

Solutions

4QuestionChoosing Appropriate Factorization MethodsAssessment Practice
4 marks~6 minCriterion C
A rectangular solar panel has an area modelled by A(x)=3x2−12A(x) = 3x^2 - 12 square metres, where xx is a design parameter in metres. Engineers require the panel to have zero net area at the boundary values of xx.

The graph of y=3x2−12y = 3x^2 - 12 crosses the xx-axis at x=−2x = -2 and x=2x = 2.
a
Show that 3x2−12=3(x−2)(x+2)3x^2 - 12 = 3(x-2)(x+2) by first extracting the common factor, then applying the difference of squares identity a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b). [2]
b
Explain how the xx-intercepts of the graph confirm that the factored form 3(x−2)(x+2)3(x-2)(x+2) correctly represents A(x)A(x). [1]
c
The engineers state: "Only positive values of xx are physically meaningful, so only the boundary value x=2x = 2 is relevant to our design." Assess whether this statement is mathematically valid. [1]
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Solutions

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