**CBSE Class 9 Maths Lab Manual – Quadratic Polynomials**

**Objective**

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

- x
^{2}+ 5x + 6 - x
^{2}– 5x + 6

**Prerequisite Knowledge**

- Knowledge of quadratic equations.
- Splitting of the middle term of a quadratic polynomial as ax
^{2}+ bx+ c = a(x+p)(x – q)

where -p + q = \(-\frac { b }{ a }\), -pq = \(\frac { c }{ a }\) - Area of a rectangle = l x b
- 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**

- Every x
^{2}represents the area of pink square of side x-units.

Therefore, to represent 2x^{2}, use two pink squares of side x units each. Take x as 3 units.

- 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)]

- 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).

- 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 x^{2} + 5x + 6 which is of the form (ax^{2} + bx + 2).

- The polynomial x
^{2}+ 5x + 6

⇒ x^{2}+ 3x + 2x + 6 can be factorized as (x+ 3)(x+2). - To present x
^{2}, draw a pink square of xunits [fig. (i) ].

- To represent 3x, draw three rectangular strips of green colour of dimension (1 x x) [fig. (ii)].

- To represent 2x, draw two green rectangular strips of dimensions (1 x x) [fig. (iii)].

- To represent 6, draw 6 yellow unit squares [fig. (iv)].

- Cut all the strips from the glazed paper.
- Now, paste all the strips together on the white sheet of paper as shown in fig.(v).

**Case II**

Consider the expression x^{2} – 5x + 6 and factorize it x^{2} – 3x – 2x + 6 = (x – 3)(x – 2).

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

- Now, paste all the five blue strips over the pink polygon as shown in fig. (vii).

**Observation and Calculation**

**Case I**

x^{2} + 5x + 6

area of 5 green strips = 5x=2x+3x

area of pink square = x^{2}

area of 6 yellow unit squares = 6

total area of rectangle obtained = x^{2} + 3x + 2x + 6 = x^{2} + 5x + 6 = (x+3)(x+2)

**Case II**

x^{2} – 5x + 6

area of 5 blue rectangular strips = 5x (negative)

area of a pink square = x^{2}

area of 6 yellow unit squares = 6

total area of pink rectangle obtained after pasting all strips

= (x – 2)(x – 3)

= x^{2} – 2x – 3x + 6

= x^{2} – 5x + 6x

∴ x^{2} – 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 ax^{2} + bx + c or ax^{2} – bx + c.

**Remarks**

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

**Activity Time**

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

- x
^{2}– x – 6 - 2x
^{2}+ 5x + 2

**Viva Voce**

**Question 1.**

How many linear factors can be in a quadratic polynomial ?

**Answer:**

2 linear factors.

**Question 2.**

Find two numbers whose sum is 1 and product is -12.

**Answer:**

-3 and 4.

**Question 3.**

Factorize: x^{2} + 7x + 12.

**Answer:**

(x + 3)(x + 4).

**Question 4.**

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

**Answer:**

√6 and -√6

**Question 5.**

What is the degree of a quadratic polynomial ?

**Answer:**

The degree of a quadratic polynomial is 2.

**Question 6.**

Give one example of a binomial.

**Answer:**

x + 5y.

**Question 7.**

Is 2 + x^{2} + x a polynomial ?

**Answer:**

Yes.

**Question 8.**

Whatis the degree of ax^{2} + bx + c?

**Answer:**

2.

**Question 9.**

Write the product of (2x – 1)(x + 1).

**Answer:**

2x^{2} + x – 1

**Question 10.**

Is y^{2} + \(\frac { 2 }{ y }\) + 5 a polynomial?

**Answer:**

No, as power of y in \(\frac { 2 }{ y }\) is -1

**Multiple Choice Questions**

**Question 1.**

Factorize the quadratic polynomial x^{2}+ 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 x^{2} – 6x + 8 :

(i) (x – 4)(x – 2)

(ii) (x – 4)(x + 2)

(iii) (x + 4)(x – 2)

(iv) none of these

**Question 3.**

Factorize x^{2} – 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) x^{2} + 3x + 2

(ii) x^{2} + 3x – 2

(iii) x^{2} – 3x + 2

(iv) none of these

**Question 5.**

What is the degree of 3x^{2} + 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 x^{2} – 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 (2x^{2} – 4) = 2x^{2} + 5x a quadratic polynomial ?

(i) yes

(ii) no

(iii) can’t say

(iv) none of these

**Answers**

- (i)
- (i)
- (ii)
- (iii)
- (iii)
- (iii)
- (i)
- (ii)
- (i)
- (ii)

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