Algebraic Simplification and Expansion
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Algebraic Simplification and Expansion — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionExpanding Double Brackets BinomialsConcept Practice
2 marks~3 minCriterion A
A garden designer plans a rectangular patio with side lengths (x+3)(x + 3) m and (x+5)(x + 5) m, where x>0x > 0. The rectangle is divided into four smaller rectangles by splitting each side at the integer term.
a
Show that the total area of the patio can be written as x2+8x+15x^2 + 8x + 15. [1]
b
The designer has 40 m² of paving material available. Given that x=2x = 2, advise the designer whether the available material is sufficient to complete the patio. [1]
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Solutions

2QuestionExpanding Double Brackets BinomialsAssessment Practice
10 marks~15 minCriterion D
A concert venue has a maximum capacity of 4000 seats. When the ticket price is 50 dollars, 2000 people are expected to attend. Market research shows that for every 2 dollar decrease in ticket price, 120 more people are expected to attend.

Let xx represent the number of 2 dollar decreases in ticket price.

Revenue is modelled by R(x)=(50−2x)(2000+120x)R(x) = (50 - 2x)(2000 + 120x).
a
Expand and simplify R(x)R(x) into the form R(x)=ax2+bx+cR(x) = ax^2 + bx + c. [2]
b
Deduce the value of xx that maximises revenue, and hence find the corresponding ticket price. [3]
c
The venue manager claims that the model predicts a maximum revenue above 105 000 dollars. Assess whether this claim is correct, and explain one reason why the model may not reflect actual revenue at this ticket price. [5]
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3QuestionCombining 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

4QuestionCombining Like Terms After ExpansionAssessment Practice
6 marks~9 minCriterion B
The product (x+1)(x+2)(x+3)⋯(x+n)(x+1)(x+2)(x+3)\cdots(x+n) generates a family of polynomials. A cryptographer uses the constant term of this product to count the number of ways to assign distinct integer keys 1,2,…,n1, 2, \ldots, n to nn slots.
a
Expand (x+1)(x+1), (x+1)(x+2)(x+1)(x+2), (x+1)(x+2)(x+3)(x+1)(x+2)(x+3), and (x+1)(x+2)(x+3)(x+4)(x+1)(x+2)(x+3)(x+4), writing each result in descending powers of xx. [2]
b
Deduce the constant term and the coefficient of xn−1x^{n-1} for each expansion in part (a), expressing both as general formulas in nn. [2]
c
Justify why the constant term always equals n!n! and the coefficient of xn−1x^{n-1} always equals 1+2+⋯+n1+2+\cdots+n, using the structure of polynomial multiplication. Then advise the cryptographer whether 55 distinct integer keys provide enough unique assignments to serve as a secure one-time code, given that a code is considered secure only if more than 100100 distinct assignments exist. [2]

Solutions

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