Algebraic Simplification and Expansion
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Algebraic Simplification and Expansion — Free MYP5 Mathematics (Standard) Practice Questions

1QuestionCombining Like Terms After ExpansionConcept Practice
4 marks~6 minCriterion C
A mobile data company advertises two pricing plans:

f(x)=2(x+3)andg(x)=2x+6f(x) = 2(x + 3) \quad \text{and} \quad g(x) = 2x + 6

where xx is the number of gigabytes used and the output is the cost in dollars.
a
Show that f(x)=2x+6f(x) = 2x + 6 by expanding using the distributive property. [1]
b
Explain why f(x)f(x) and g(x)g(x) represent the same function for all values of xx. [1]
c
A customer claims that Plan ff is cheaper than Plan gg. Advise the customer whether this claim is correct, and what it means for their choice between the two plans. [2]
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Solutions

2QuestionCombining Like Terms After ExpansionAssessment Practice
4 marks~6 minCriterion D
A small business owner models monthly profit (in dollars) as P(x)=(x+50)(200−2x)P(x) = (x + 50)(200 - 2x), where xx is the number of one-dollar price increases applied to the product. The model assumes demand decreases linearly as price rises.
a
Expand P(x)=(x+50)(200−2x)P(x) = (x + 50)(200 - 2x) and write it in the form ax2+bx+cax^2 + bx + c, stating the values of aa, bb, and cc. [2]
b
Advise the business owner whether this model is a sufficient basis for determining an optimal pricing strategy. Refer to both the mathematical structure of P(x)P(x) and real-world factors the model does not capture. [2]
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Solutions

3QuestionExpanding Double Brackets BinomialsConcept Practice
4 marks~6 minCriterion A
A farmer models the area of a rectangular field (in square metres) using the expression (x+5)(x+3)(x+5)(x+3), where xx is a positive side-length in metres.
a
Expand and simplify (x+5)(x+3)(x+5)(x+3). [2]
b
The farmer needs a field with an area of at least 77 m277 \text{ m}^2. Given that x=7x = 7, justify whether this field meets the requirement, using your expression from part (a). [1]
c
Discuss one strength and one limitation of using this algebraic expression as a model for planning crop planting. [1]
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Solutions

4QuestionCombining Like Terms After ExpansionAssessment Practice
5 marks~8 minCriterion B
Consider the following expansions:

(x+1)(x+2)=x2+3x+2(x+1)(x+2) = x^2 + 3x + 2

(x+2)(x+3)=x2+5x+6(x+2)(x+3) = x^2 + 5x + 6

(x+3)(x+4)=x2+7x+12(x+3)(x+4) = x^2 + 7x + 12

(x+4)(x+5)=x2+9x+20(x+4)(x+5) = x^2 + 9x + 20
a
Using the pattern above, predict:

(i) the constant term in the expansion of (x+5)(x+6)(x+5)(x+6). [1]

(ii) the coefficient of xx in the expansion of (x+5)(x+6)(x+5)(x+6). [1]
b
Deduce a general formula for (x+a)(x+b)(x+a)(x+b), showing full algebraic working. [2]
c
Apply your formula to expand (x+10)(x+15)(x+10)(x+15). [1]
d
A student claims: "The coefficient of xx is always equal to the product of the two constants." Evaluate this claim using your general formula and one example from the table above. [2]

(e) A rectangular garden has side lengths (x+10)(x + 10) metres and (x+15)(x + 15) metres. The designer states the garden can only be built if its area exceeds 500 m2500\ \text{m}^2 when x=10x = 10. Justify whether the designer's condition is met. [1]

Solutions

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