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

1QuestionFactoring Difference of SquaresConcept Practice
4 marks~6 minCriterion A
A landscape architect models a rectangular garden bed. Its area in square metres is described by the function y=x2−9y = x^2 - 9, where xx is a design parameter.
a
Deduce the factored form of x2−9x^2 - 9. [1]
b
Using your answer to part (a), determine the values of xx where y=0y = 0. [2]
c
The design parameter xx must be positive. Justify which x-intercept is valid for this context. [1]
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2QuestionFactoring Difference of SquaresAssessment Practice
4 marks~6 minCriterion D
A landscape designer plans a square garden with side length a=10a = 10 m. A square pond with side length b=2b = 2 m will be placed in one corner. The remaining garden area (excluding the pond) is modelled using the identity a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b).
a
Show that the area of the garden excluding the pond is 96 m296 \text{ m}^2, using the difference of squares identity. [1]
b
The pond's side length is remeasured as 2.12.1 m. Calculate the revised area and deduce the percentage change from the original result. [1]
c
The designer considers replacing the square pond with a circular pond of diameter 22 m. Advise the designer whether the difference of squares identity remains a valid model and whether the area calculation must be adjusted. [2]
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3QuestionChoosing 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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4QuestionFactoring Difference of SquaresAssessment Practice
5 marks~8 minCriterion B
A tile manufacturer cuts square ceramic tiles of side length aa cm, then removes a smaller square of side length bb cm from one corner to create an L-shaped tile for border edging.
a
Show that (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2 by expanding the left-hand side algebraically. State the property used at each step. [2]
b
Construct an area model for the L-shaped tile. Show that the remaining area can be expressed both as a2−b2a^2 - b^2 and as (a+b)(a−b)(a+b)(a-b). [2]
c
A tile is described as "large" if its L-shaped area exceeds 300 cm2300\ \text{cm}^2. Given a=20a = 20 and b=5b = 5, justify whether this tile qualifies as large. [1]
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