## CBSE Class 9 Maths Lab Manual – Quadratic Polynomials

Objective
Learning geometrical representation of the factorization of the following quadratic polynomials:

1. x2 + 5x + 6
2. x2 – 5x + 6

Prerequisite Knowledge

2. Splitting of the middle term of a quadratic polynomial as ax2 + bx+ c = a(x+p)(x – q)
where -p + q = $$-\frac { b }{ a }$$, -pq = $$\frac { c }{ a }$$
3. Area of a rectangle = l x b
4. Area of a square = (side)2

Materials Required
Glazed papers (blue, green, orange, yellow and pink), white sheet of paper, geometry box, ruler, pair of scissors and gluestick.

Procedure

1. Every x2 represents the area of pink square of side x-units.
Therefore, to represent 2x2, use two pink squares of side x units each. Take x as 3 units.
2. Every x represents area of a green rectangular strip of dimensions (1 x x).
[For 5x, use 5 green strips each of dimensions (1 x x)]
3. Every (-x) is represented by a blue rectangular strip of dimensions (1 x x).
For (-5x), use 5 blue strips each of dimensions (1 x x).
4. All positive integers are represented by yellow unit squares and all negative integers are shown by orange squares.

Case I
Let us consider the expression x2 + 5x + 6 which is of the form (ax2 + bx + 2).

1. The polynomial x2 + 5x + 6
⇒ x2 + 3x + 2x + 6 can be factorized as (x+ 3)(x+2).
2. To present x2, draw a pink square of xunits [fig. (i) ].
3. To represent 3x, draw three rectangular strips of green colour of dimension (1 x x) [fig. (ii)].
4. To represent 2x, draw two green rectangular strips of dimensions (1 x x) [fig. (iii)].
5. To represent 6, draw 6 yellow unit squares [fig. (iv)].
6. Cut all the strips from the glazed paper.
7. Now, paste all the strips together on the white sheet of paper as shown in fig.(v).

Case II
Consider the expression x2 – 5x + 6 and factorize it x2 – 3x – 2x + 6 = (x – 3)(x – 2).

1. Cut a pink square of dimension xunits (say 8 units).
2. To represent 6, cut six yellow squares of dimension 1 unit.
3. To represent -5x {(-3x) + (-2x)}, cut five blue strips of dimension (1 x x).
4. Paste the pink square strips and all the yellow squares on a white sheet paper as shown in fig. (vi).
5. Now, paste all the five blue strips over the pink polygon as shown in fig. (vii).

Observation and Calculation
Case I
x2 + 5x + 6
area of 5 green strips = 5x=2x+3x
area of pink square = x2
area of 6 yellow unit squares = 6
total area of rectangle obtained = x2 + 3x + 2x + 6 = x2 + 5x + 6 = (x+3)(x+2)

Case II
x2 – 5x + 6
area of 5 blue rectangular strips = 5x (negative)
area of a pink square = x2
area of 6 yellow unit squares = 6
total area of pink rectangle obtained after pasting all strips
= (x – 2)(x – 3)
= x2 – 2x – 3x + 6
= x2 – 5x + 6x
∴ x2 – 5x + 6 = (x – 3)(x – 2)

Result
We verified the factors of two quadratic polynomials geometrically by paper cutting and pasting.

Learning Outcome
Above method gives us the geometrical interpretation of the factorization of quadratic expressions of the form ax2 + bx + c or ax2 – bx + c.

Remarks

1. Pasting of blue strips over pink area means reducing pink area.
2. The pink portion so obtained represents the factors of the given quadratic expression.
3. Students may take different colour combinations.

Activity Time
By using paper cutting and pasting method, represent the factors of following quadratic expressions:

1. x2 – x – 6
2. 2x2 + 5x + 2

Viva Voce

Question 1.
How many linear factors can be in a quadratic polynomial ?
2 linear factors.

Question 2.
Find two numbers whose sum is 1 and product is -12.
-3 and 4.

Question 3.
Factorize: x2 + 7x + 12.
(x + 3)(x + 4).

Question 4.
Find two numbers whose sum is 0 and product is -6.
√6 and -√6

Question 5.
What is the degree of a quadratic polynomial ?
The degree of a quadratic polynomial is 2.

Question 6.
Give one example of a binomial.
x + 5y.

Question 7.
Is 2 + x2 + x a polynomial ?
Yes.

Question 8.
Whatis the degree of ax2 + bx + c?
2.

Question 9.
Write the product of (2x – 1)(x + 1).
2x2 + x – 1

Question 10.
Is y2 + $$\frac { 2 }{ y }$$ + 5 a polynomial?
No, as power of y in $$\frac { 2 }{ y }$$ is -1

Multiple Choice Questions

Question 1.
Factorize the quadratic polynomial x2+ 6x + 8 :
(i) (x + 4)(x + 2)
(ii) (x – 4)(x – 2)
(iit) (x + 5)(x + 3)
(iv) none of these

Question 2.
Write the factors of x2 – 6x + 8 :
(i) (x – 4)(x – 2)
(ii) (x – 4)(x + 2)
(iii) (x + 4)(x – 2)
(iv) none of these

Question 3.
Factorize x2 – 5x + 6
(i) (x – 3)(x + 2)
(ii) (x – 3)(x – 2)
(iii) (x + 3)(x + 2)
(iv) none of these

Question 4.
Write the quadratic polynomial for (x— 1) (x— 2)
(i) x2 + 3x + 2
(ii) x2 + 3x – 2
(iii) x2 – 3x + 2
(iv) none of these

Question 5.
What is the degree of 3x2 + 2x + 1 ?
(i) 1
(ii) 3
(iii) 2
(iv) none of these

Question 6.
What will be the degree of a biquadratic polynomial ?
(i) 2
(ii) 3
(iii) 4
(iv) none of these

Question 7.
How many zeroes , are possible of a quadratic polynomial ?
(i) 2
(ii) 3
(iii) 1
(iv) none of these

Question 8.
Is x – 2 a factor of x2 – 6x + 5 ?
(i) yes
(ii) no
(iii) can’t say
(iv) none of these

Question 9.
Is (x – 4)(x – 5) = 7x a quadratic polynomial ?
(i) yes
(ii) no
(iii) linear
(iv) none of these

Question 10.
Is (2x2 – 4) = 2x2 + 5x a quadratic polynomial ?
(i) yes
(ii) no
(iii) can’t say
(iv) none of these