Literal calculation: math exercises in 3rd grade corrected in PDF.

A series of exercises on remarkable identities and literal calculus in the third grade (3ème).These exercises are intended for teachers and students of the third grade who wish to revise the chapter on remarkable identities and literal calculus online.

To continue the notions seen in fifth grade (simple distributivity) and in fourth grade (double distributivity),

you will find in this series of mathematical exercises on literal calculation, the following notions :

  • definition of an algebraic or literal expression;
  • reduce and expand a literal expression;
  • factor a literal expression;
  • substitution.

These corrected exercises of maths in third grade (3ème) have been written by a maths teacher

and can be viewed online or downloaded in PDF format.

Exercise 1:

Expand and reduce each expression.


Exercise 2:

Expand and reduce the following expression:


Exercise 3:

In each case, only one answer is correct.
Copy the correct answer.

a. If we expand and reduce the expression (x + 2)(3x-1),
we obtain :

3x^2+5x-2 or 3x^2+6x+2 or 3x^2-1

b. The expanded form of (x-1)^2 is:

(x-1)(x+1) or x^2-2x+1 or x^2+2x+1.
c. A factorized expression of (x\,-\,1)^2\,-\,16 is:
(x+3)(x-5) or (x+4)(x-4) or x^2-2x-15.

d. A factorized expression of x^2\,-\,36 is :
(x-6)^2 or (x+18)(x-18) or (x-6)(x+6).

Exercise 4:

a. Give the result provided by the calculation program if we choose as
number of departures :

-2; 5 then 10.

b. Show that the result is always the square of a whole number.

Calculation program

Exercise 5:

The unit of length is the centimeter.
x denotes a number (x > 1).
a. For what value of x is the perimeter of the quadrilateral QUAD 32 cm ?
b. What is then the quadrilateral nature of QUAD?

Quadrilateral and literal calculation

Exercise 6:

AENT is a square with a perimeter of 56 cm.

PAE is an isosceles triangle at P.
a. Calculate AE.

b. For what length of [AP] is the perimeter of the pentagon PENTA equal to 60 cm? Justify.

Square and triangle

Exercise 7:

Expand the following literal expressions:








Exercise 8:

x denotes a number greater than or equal to 2.
ABCD is a square and ABEF is a rectangle.

1. Express as a function of x;
a. the length AD ;

b. the area A of the square ABCD ;

c. the area B of rectangle ABEF ;
d. the area ‘C of the ECDF rectangle.

2. a. Express the areas B and C and their sum in expanded and reduced form.

b. Check that this sum is equal to A.

rectangle, square and literal calculation

Exercise 9:

Here are two calculation programs.

Calculation programs

a. Apply each program to the numbers :

3; 10 and – 5 and then to another randomly chosen number
What do we see? Make a conjecture.

b. We note n the number chosen at the beginning.

Express the result obtained with each program as a function of n.

Prove the conjecture made in question a.

Exercise 10:

x is a positive number.
Here is a rectangle with sides of varying lengths.
a. Léa has built the program below with the Scratch software.

Calculation program with scratch

What do the variables l and L represent?

b. What is the role of Lea’s program?

c. Lea says:”P=3x+9;A=x^2+7x+\,10.”
Is she right? Explain.

d. Carry out this program. Test it by giving x the value 3, then the value 10.

Exercise 11:

Associate each expression on the left with its factorized form on the right.

Factor and expand in literal calculation

Exercise 12:

Consider the expression C=(3x-1)^2-(3x-1)(2x+3).
1) Expand and reduce C .
2) Factorize C .
3) Solve the equation: (3x-1)(x-4)=0.

4) Calculate C for x=2.

Exercise 13:

Develop using the indicated model.
Square of a sum | Square of a difference
(a+b)^2=a^2+2ab+b^2 |(a-b)^2=a^2-2ab+b^2

Exercise 14:

We know that multiplying the sum of two numbers by their difference gives :

I=(x+8)(x-8) and J=(t-5)(t+5).

Exercise 15:

Factor each expression with a remarkable identity.

Exercise 16:

Recognize a difference of two squares in each expression, then factor.

Exercise 17:

Reduce each expression using a remarkable identity.



Exercise 18:

Expand and collapse each expression.


Exercise 19:

x is a relative number.
Using remarkable identities, copy and complete the table below.

Factored and expanded forms

Exercise 20:

Copy and complete using a remarkable identity.
a. 4x^2+12x+9=(...\,+...)^2
b. 16x^2-\,40x+25=(...-...)^2
c. 9x^2-64=(...+...)(...-...)

Exercise 21:

QCm on literal calculation

Exercise 22:

  1. Recall the three remarkable identities.
  2. We want to develop (6x\,+\,5)^2:
    1. Which one will we use? Then specify the value of a and b.
    2. What is the development of (6x\,+\,5)^2?

Exercise 23:

Complete and finish developments:

a. (x\,-\,4)^2\,=\,....\,^2\,-\,2\times  \,....\,\times  \,....\,+....\,^2;

b. (3x+2)^2\,=\,....\,^2\,+\,2\times  \,....\,\times  \,....\,+....\,^2

Exercise 24:

Same exercise as the previous one.

a. (x-\frac{1}{2})^2\,=\,....\,^2\,-\,2\times  \,....\,\times  \,....\,+....\,^2

b. (\frac{3}{5}x+\frac{7}{3})^2\,=\,....\,^2\,+\,2\times  \,....\,\times  \,....\,+....\,^2

Exercise 25:



Exercise 26:



Exercise 27:

Expand then reduce :


Exercise 28:

State the factorized form of these expanded remarkable identities:


Exercise 29:

Factor the following expressions:


Exercise 30:

1) Expand then reduce D=(a+5)^2-(a-5)^2.

2) We pose D\,=\,10\,005^2-\,9\,995^2.

3) Without using the calculator and using question 1, find the value of D.

Exercise 31:

We give E=(3x-5)(2x+1)-(3x-5)^2.

1) Expand and reduce E.

2) Factorize E.

3) Expand the expression obtained in question 2.

What is the result

Exercise 32:

We give E=(2x+3)^2-16.

1) Show that E can be written 4x^2\,+\,12x\,-\,7.

2) Calculate E for: x\,=\,2; x\,=-3.

3) Factor E. Expand the resulting expression.

What is the result?

Cette publication est également disponible en : Français (French) العربية (Arabic)

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