CBSE Class 9 Mathematics Worksheet - Polynomials | PDF - Free Printable

Let's solve each question from the Maths Worksheet step-by-step, with clear explanations.

---

Q01: Factorize by splitting the middle term

#### a) $ 3x^2 + 19x + 30 $

We need to split the middle term $19x$ into two terms whose product is $3 \times 30 = 90$ and sum is $19$.

Find two numbers:
- Factors of 90: $ (1,90), (2,45), (3,30), (5,18), (6,15), (9,10) $
- $9 + 10 = 19$, and $9 \times 10 = 90$

So:
$$
3x^2 + 19x + 30 = 3x^2 + 9x + 10x + 30
= 3x(x + 3) + 10(x + 3)
= (3x + 10)(x + 3)
$$

✔ Answer: $ (3x + 10)(x + 3) $

---

#### b) $ 2\sqrt{2}x^2 + 9x + 5\sqrt{2} $

Multiply and factor using the same method.
Product: $ 2\sqrt{2} \times 5\sqrt{2} = 2 \times 5 \times (\sqrt{2})^2 = 10 \times 2 = 20 $
We need two numbers that multiply to 20 and add to 9

Try: $4$ and $5$: $4+5=9$, $4\times5=20$

Now split $9x$ as $4x + 5x$:

$$
2\sqrt{2}x^2 + 4x + 5x + 5\sqrt{2}
= 2x(\sqrt{2}x + 2) + 5(\sqrt{2}x + 2)
= (2x + 5)(\sqrt{2}x + 2)
$$

✔ Answer: $ (2x + 5)(\sqrt{2}x + 2) $

---

#### c) $ 4x^2 - 13x + 10 $

Product: $4 \times 10 = 40$
Need two numbers that multiply to 40 and add to $-13$

Try: $-8$ and $-5$: $-8 + (-5) = -13$, $(-8)(-5)=40$

Split:
$$
4x^2 - 8x - 5x + 10
= 4x(x - 2) -5(x - 2)
= (4x - 5)(x - 2)
$$

✔ Answer: $ (4x - 5)(x - 2) $

---

Q02: Factorize by Factor Theorem

The Factor Theorem: If $f(a) = 0$, then $(x - a)$ is a factor.

#### a) $ x^3 + 9x^2 + 23x + 15 $

Try rational roots: factors of 15 over 1 → $\pm1, \pm3, \pm5, \pm15$

Try $x = -1$:
$(-1)^3 + 9(-1)^2 + 23(-1) + 15 = -1 + 9 - 23 + 15 = 0$ ✔ So $(x + 1)$ is a factor.

Now divide polynomial by $(x + 1)$ using synthetic division:

```
-1 | 1 9 23 15
| -1 -8 -15
-------------------
1 8 15 0
```

Quotient: $x^2 + 8x + 15$

Factor: $x^2 + 8x + 15 = (x + 3)(x + 5)$

So full factorization: $ (x + 1)(x + 3)(x + 5) $

✔ Answer: $ (x + 1)(x + 3)(x + 5) $

---

#### b) $ x^3 + 6x^2 + 11x + 6 $

Try $x = -1$:
$-1 + 6 - 11 + 6 = 0$ ✔ So $(x + 1)$ is a factor.

Synthetic division:

```
-1 | 1 6 11 6
| -1 -5 -6
------------------
1 5 6 0
```

Quotient: $x^2 + 5x + 6 = (x + 2)(x + 3)$

So: $ (x + 1)(x + 2)(x + 3) $

✔ Answer: $ (x + 1)(x + 2)(x + 3) $

---

Q03: Factorize using suitable identity

#### a) $ 4x^2 + 12xy + 9y^2 $

Recognize this as: $ (2x)^2 + 2(2x)(3y) + (3y)^2 = (2x + 3y)^2 $

✔ Answer: $ (2x + 3y)^2 $

---

#### b) $ 2a^5 - 54a^2 $

Factor out common term: $2a^2(a^3 - 27)$

Now $a^3 - 27 = a^3 - 3^3 = (a - 3)(a^2 + 3a + 9)$

So:
$$
2a^2(a - 3)(a^2 + 3a + 9)
$$

✔ Answer: $ 2a^2(a - 3)(a^2 + 3a + 9) $

---

#### c) $ 2\sqrt{2}x^3 + 3\sqrt{3}y^3 $

This is sum of cubes: $A^3 + B^3 = (A + B)(A^2 - AB + B^2)$

But here we have coefficients.

Write as:
$$
( \sqrt{2}x )^3 \cdot 2 / \sqrt{2} ??? \quad \text{Wait — better: factor directly.}
$$

Actually:
$$
2\sqrt{2}x^3 = (\sqrt{2}x)^3 \cdot \frac{2}{\sqrt{2}}? \quad \text{Not helpful.}
$$

Alternatively, note:
$$
2\sqrt{2}x^3 = (\sqrt{2}x)^3 \cdot \frac{2}{\sqrt{2}} = (\sqrt{2}x)^3 \cdot \sqrt{2} \quad \text{No.}
$$

Wait: Try factoring as:
$$
= (\sqrt{2}x)^3 \cdot \sqrt{2} + (\sqrt{3}y)^3 \cdot \sqrt{3} \quad \text{Still messy.}
$$

Better idea: Let’s write it as:
$$
= (\sqrt{2}x)^3 \cdot \sqrt{2} + (\sqrt{3}y)^3 \cdot \sqrt{3} \quad \text{Not useful.}
$$

Actually, it's not a standard identity unless we factor out something.

But notice:
$2\sqrt{2}x^3 = (\sqrt{2}x)^3 \cdot \sqrt{2}$ — no.

Wait! Perhaps it's meant to be interpreted differently.

But actually, this expression does not factor nicely using standard identities unless it's written as a sum of cubes.

Let’s suppose:
Let $ A = \sqrt{2}x $, $ B = \sqrt{3}y $

Then $ A^3 = 2\sqrt{2}x^3 $? No: $ (\sqrt{2}x)^3 = 2\sqrt{2}x^3 $ ✔ Yes!

Similarly: $ (\sqrt{3}y)^3 = 3\sqrt{3}y^3 $ ✔

So:
$$
2\sqrt{2}x^3 + 3\sqrt{3}y^3 = (\sqrt{2}x)^3 + (\sqrt{3}y)^3
$$

Use sum of cubes identity:
$$
A^3 + B^3 = (A + B)(A^2 - AB + B^2)
$$

So:
$$
= (\sqrt{2}x + \sqrt{3}y)\left[ (\sqrt{2}x)^2 - (\sqrt{2}x)(\sqrt{3}y) + (\sqrt{3}y)^2 \right]
= (\sqrt{2}x + \sqrt{3}y)(2x^2 - \sqrt{6}xy + 3y^2)
$$

✔ Answer: $ (\sqrt{2}x + \sqrt{3}y)(2x^2 - \sqrt{6}xy + 3y^2) $

---

#### d) $ x^5 - x $

Factor out $x$:
$$
x(x^4 - 1)
= x(x^2 - 1)(x^2 + 1)
= x(x - 1)(x + 1)(x^2 + 1)
$$

✔ Answer: $ x(x - 1)(x + 1)(x^2 + 1) $

---

#### e) $ x^6 - y^6 $

This is difference of squares:
$$
= (x^3)^2 - (y^3)^2 = (x^3 - y^3)(x^3 + y^3)
$$

Now use sum and difference of cubes:
- $x^3 - y^3 = (x - y)(x^2 + xy + y^2)$
- $x^3 + y^3 = (x + y)(x^2 - xy + y^2)$

So:
$$
x^6 - y^6 = (x - y)(x^2 + xy + y^2)(x + y)(x^2 - xy + y^2)
$$

Or rearranged:
$$
= (x - y)(x + y)(x^2 + xy + y^2)(x^2 - xy + y^2)
$$

✔ Answer: $ (x - y)(x + y)(x^2 + xy + y^2)(x^2 - xy + y^2) $

---

#### f) $ (a - b)^3 + (b - c)^3 + (c - a)^3 $

Use identity:
If $x + y + z = 0$, then $x^3 + y^3 + z^3 = 3xyz$

Let:
- $x = a - b$
- $y = b - c$
- $z = c - a$

Then $x + y + z = (a - b) + (b - c) + (c - a) = 0$

So:
$$
x^3 + y^3 + z^3 = 3xyz = 3(a - b)(b - c)(c - a)
$$

✔ Answer: $ 3(a - b)(b - c)(c - a) $

---

#### g) $ x^8 - y^8 $

Difference of squares:
$$
= (x^4)^2 - (y^4)^2 = (x^4 - y^4)(x^4 + y^4)
$$

Now:
- $x^4 - y^4 = (x^2)^2 - (y^2)^2 = (x^2 - y^2)(x^2 + y^2) = (x - y)(x + y)(x^2 + y^2)$

So:
$$
x^8 - y^8 = (x - y)(x + y)(x^2 + y^2)(x^4 + y^4)
$$

✔ Answer: $ (x - y)(x + y)(x^2 + y^2)(x^4 + y^4) $

---

#### h) $ 27x^3 - 135x^2 + 225x - 125 $

Notice all coefficients divisible by 27? Try factoring.

Check if it's a perfect cube?

Let’s try:
$$
(3x)^3 = 27x^3 \\
(5)^3 = 125
$$

So maybe $ (3x - 5)^3 $?

Compute:
$$
(3x - 5)^3 = 27x^3 - 3(9x^2)(5) + 3(3x)(25) - 125 = 27x^3 - 135x^2 + 225x - 125
$$

Yes! Exactly matches.

✔ Answer: $ (3x - 5)^3 $

---

Q04: Evaluate using identity

#### a) $ 998^3 $

Note: $998 = 1000 - 2$

Use identity: $ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 $

Let $a = 1000$, $b = 2$

$$
(1000 - 2)^3 = 1000^3 - 3(1000)^2(2) + 3(1000)(4) - 8
= 10^9 - 6 \times 10^6 + 12000 - 8
= 1000000000 - 6000000 + 12000 - 8
= 994011992
$$

✔ Answer: $ 994011992 $

---

#### b) $ 10.2^3 $

$10.2 = 10 + 0.2$

Use: $ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 $

$a = 10$, $b = 0.2$

$$
= 1000 + 3(100)(0.2) + 3(10)(0.04) + 0.008
= 1000 + 60 + 1.2 + 0.008 = 1061.208
$$

✔ Answer: $ 1061.208 $

---

#### c) $ 998^2 - 4 $

$998^2 = (1000 - 2)^2 = 1000000 - 4000 + 4 = 996004$

Then: $996004 - 4 = 996000$

Or use identity: $a^2 - b^2 = (a - b)(a + b)$

Here: $998^2 - 2^2 = (998 - 2)(998 + 2) = 996 \times 1000 = 996000$

✔ Answer: $ 996000 $

---

#### d) $ 999^2 - 1 $

Again: $a^2 - b^2 = (a - b)(a + b)$

$999^2 - 1^2 = (999 - 1)(999 + 1) = 998 \times 1000 = 998000$

✔ Answer: $ 998000 $

---

#### e) $ (-25)^3 + 10^3 + 15^3 $

Note: $-25 + 10 + 15 = 0$

So use identity: If $a + b + c = 0$, then $a^3 + b^3 + c^3 = 3abc$

So:
$$
(-25)^3 + 10^3 + 15^3 = 3(-25)(10)(15) = 3 \times (-3750) = -11250
$$

✔ Answer: $ -11250 $

---

#### f) $ 10.2 \times 9.8 $

Use: $ (a + b)(a - b) = a^2 - b^2$

$10.2 = 10 + 0.2$, $9.8 = 10 - 0.2$

So:
$$
(10 + 0.2)(10 - 0.2) = 10^2 - (0.2)^2 = 100 - 0.04 = 99.96
$$

✔ Answer: $ 99.96 $

---

Q05: If 5 is a zero of $x^3 + kx^2 + 2x + 8$, find $k$

Since 5 is a root, plug in $x = 5$:

$$
(5)^3 + k(5)^2 + 2(5) + 8 = 0 \\
125 + 25k + 10 + 8 = 0 \\
143 + 25k = 0 \\
25k = -143 \\
k = -\frac{143}{25} = -5.72
$$

✔ Answer: $ k = -\frac{143}{25} $

---

Q06: If $(x - 2)$ is a zero of $x^3 - 4x^2 + kx - 8$, find $k$

Plug in $x = 2$:

$$
(2)^3 - 4(2)^2 + k(2) - 8 = 0 \\
8 - 16 + 2k - 8 = 0 \\
-16 + 2k = 0 \\
2k = 16 \\
k = 8
$$

✔ Answer: $ k = 8 $

---

Q07: If $(x - 2)$ and $(x + 3)$ are factors of $x^3 + ax^2 + bx - 30$, find $a$ and $b$

Since both are factors, $x = 2$ and $x = -3$ are roots.

Plug $x = 2$:

$$
(2)^3 + a(2)^2 + b(2) - 30 = 0 \\
8 + 4a + 2b - 30 = 0 \\
4a + 2b = 22 \quad \text{(1)}
$$

Divide by 2: $2a + b = 11$ → Equation (i)

Plug $x = -3$:

$$
(-3)^3 + a(-3)^2 + b(-3) - 30 = 0 \\
-27 + 9a - 3b - 30 = 0 \\
9a - 3b = 57 \quad \text{(2)}
$$

Divide by 3: $3a - b = 19$ → Equation (ii)

Now solve:
From (i): $b = 11 - 2a$

Substitute into (ii):
$$
3a - (11 - 2a) = 19 \\
3a - 11 + 2a = 19 \\
5a = 30 \\
a = 6
$$

Then $b = 11 - 2(6) = 11 - 12 = -1$

✔ Answer: $ a = 6, b = -1 $

---

## ✔ Final Answers Summary

Q01:

a) $ (3x + 10)(x + 3) $
b) $ (2x + 5)(\sqrt{2}x + 2) $
c) $ (4x - 5)(x - 2) $

Q02:

a) $ (x + 1)(x + 3)(x + 5) $
b) $ (x + 1)(x + 2)(x + 3) $

Q03:

a) $ (2x + 3y)^2 $
b) $ 2a^2(a - 3)(a^2 + 3a + 9) $
c) $ (\sqrt{2}x + \sqrt{3}y)(2x^2 - \sqrt{6}xy + 3y^2) $
d) $ x(x - 1)(x + 1)(x^2 + 1) $
e) $ (x - y)(x + y)(x^2 + xy + y^2)(x^2 - xy + y^2) $
f) $ 3(a - b)(b - c)(c - a) $
g) $ (x - y)(x + y)(x^2 + y^2)(x^4 + y^4) $
h) $ (3x - 5)^3 $

Q04:

a) $ 994011992 $
b) $ 1061.208 $
c) $ 996000 $
d) $ 998000 $
e) $ -11250 $
f) $ 99.96 $

Q05: $ k = -\frac{143}{25} $

Q06: $ k = 8 $

Q07: $ a = 6, b = -1 $

Let me know if you'd like these explained further or formatted for printing!