Factorising expressionsIB MYP Maths Extended: Subtopic test

1. 1
   A rectangular banner has area $(x^2+9x+20)$ square centimetres. Its length and width are expressions in $x$ of the form $(x+p)$ and $(x+q)$.

   (a)

   Factorise $x^2+9x+20$.

   [1 mark]

   • A$(x+2)(x+10)$
   • B$(x+4)(x+5)$
   • C$(x-4)(x-5)$
   • D$(x+1)(x+20)$

   (b)

   When $x=3$ the area of the banner is $56$ cm$^2$. Use the factorised form to find the perimeter of the banner.

   [1 mark]

   • A$15$ cm
   • B$56$ cm
   • C$16$ cm
   • D$30$ cm

   (c)

   A second banner has area $(x^2+9x)$ cm$^2$. Factorise this expression, and hence find the width of this banner when its length is $(x+9)$ cm.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A rectangular tile has area $(x^2-49)$ square centimetres.

   (a)

   Factorise $x^2-49$.

   [1 mark]

   • A$(x-7)^2$
   • B$(x+7)^2$
   • C$(x-7)(x+7)$
   • D$x(x-49)$

   (b)

   The same method can be used with numbers. Use factorising to evaluate $53^2-47^2$ without a calculator.

   [1 mark]

   • A$600$
   • B$6$
   • C$100$
   • D$36$

   (c)

   Factorise fully $3x^2-75$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A rectangular vegetable bed has area $(x^2+3x-10)$ square metres and length $(x+5)$ metres, where $x>2$.

   (a)

   Factorise $x^2+3x-10$ and hence write down an expression for the width of the bed.

   [3 marks]

   (b)

   The area of the bed is $18$ square metres. Form an equation, solve it by factorising, and find the length and width of the bed.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A square courtyard has side $x$ metres. It is rebuilt as a rectangle whose area is $(x^2-9)$ square metres.

   (a)

   (i) Factorise $x^2-9$.
   (ii) The original square has side $10$ m. Find the dimensions and the area of the rectangle.
   (iii) Show that the rectangle's area is $9$ m$^2$ less than the square's for any value of $x$.

   [6 marks]

   (b)

   A different courtyard of side $x$ metres is rebuilt by making one side $k$ metres shorter and the other side $k$ metres longer, so the new area is $(x^2-k^2)$ square metres.
   (i) Factorise $x^2-k^2$.
   (ii) When $x=20$ the new area is $336$ m$^2$. Find $k$.
   (iii) State the dimensions of this rectangle, and explain why $k$ must be less than $20$.

   [6 marks]

   Total for question 4: 12 marks

End of questions