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QUESTION: 1

What are the factors of ax+by+bx+az+ay+bz?

Solution:

ax+by+bx+az+ay+bz, rearranging we get ax+ay+az+bx+by+bz

a(x+y+z)+b(x+y+z)=(a+b)(x+y+z). Hence the factors are (a+b), (x+y+z)

QUESTION: 2

Which of the following is one of the factors of x^{4}+4?

Solution:

x⁴ + 4

= (x² + 2)² - 4x²

= (x² + 2)² - (2x)²

a² - b² = (a + b)(a - b)

= (x² + 2 + 2x)(x² + 2 - 2x)

QUESTION: 3

What are the factors of x^{4}+2x^{2}+9?

Solution:

Given equation is x^{4} + 2x^{2} + 9

We can rewrite this as,

(x^{2})^{2} + 6x^{2} + 9 − 4x^{2}

⇒ (x^{2} + 3)^{2} − (2x)^{2}

....Since a^{2} + 2ab + b^{2} = (a+b)^{2}

⇒ x^{4} + 2x^{2} + 9 = (x^{2} − 2x + 3)(x^{2} + 2x + 3)

....Since a^{2 }− b^{2} = (a+b)(a−b)

QUESTION: 4

For x^{2}+2x+5 to be a factor of x^{4}+ px^{2}+q, what must the respective values of p and q be?

Solution:

Let the other factor be x^{2}+ax+b. We have (x^{2}+2x+5)(x^{2}+ax+b)

= x^{4}+px^{2}+q

x^{4}(2+a)x^{3}(2a+b+5)x^{2}(5a+2b)x+5b = x4+px^{2}+q

Comparing the coefficients of corresponding terms, we get 2a+b+5 = p ......(1)

5b = q ......(2)

2+a = 0 ⇒ a =−2

5a+2b = 0 ⇒ b = 5

∴ p = 2a+b+5 = 2(−2)+5+5 = 6

q = 5b = 5(5) = 25

QUESTION: 5

What are the factors of x^{2}+xy−2xz−2yz?

Solution:

QUESTION: 6

Amrit and Pankaj expanded (x−5)^{2}. Amrit's answer is x^{2}−25 and Pankaj's answer is x^{2}−10x+25. Which of the following statements is correct?

Solution:

(x−5)^{2} = x^{2}−10x+25 using

(a−b)^{2} = a^{2}−2ab+b^{2}. So, Pankaj's answer is correct.

QUESTION: 7

Find the quotient when 5a^{2}b^{2}c^{2} is divided by 15abc.

Solution:

QUESTION: 8

Which of the following statements is correct?

Solution:

(a−4)(a−2) = a^{2}−4a−2a+8 = a^{2}−6a+8 So, the statement in option (A) is correct.

QUESTION: 9

What are the factors of x^{4}+y^{4}+x^{2}y^{2}?

Solution:

x^{4}+y^{4}+x^{2}y^{2} =(x^{4}+y^{4}+2x^{2}y^{2})−x^{2}y^{2} =(x^{2}+y^{2})^{2}−(xy)^{2}

=(x^{2}+y^{2}+xy)(x^{2}+y^{2}−xy)

QUESTION: 10

Choose the factors of 15x^{2}−26x+8 from the following.

Solution:

15x^{2}−26x+8 =15x^{2}−20x−6x+8 =5x(3x−4)−2(3x−4) =(3x−4)(5x−2)

QUESTION: 11

How many factors does (x^{9}−x) have?

Solution:

QUESTION: 12

Which of the following are the factors of

Solution:

QUESTION: 13

What is the coefficient of 'a' when 9a^{2}+18a is divided by (a+2)?

Solution:

QUESTION: 14

From the following, which are the factors of a^{2}+b−ab−a?

Solution:

QUESTION: 15

The expression (p^{2}+7p+10) is factorized and then divided by (p+5). What is the quotient?

Solution:

QUESTION: 16

Which is the correct statement in the following?

Solution:

n(3n+2) = 3n^{2}+2n

QUESTION: 17

If (x^{2}+3x+5)(x^{2}−3x+5) = m^{2}−n^{2}, what is the value of m?

Solution:

(x^{2}+3x+5)(x^{2}−3x+5) = (x^{2}+5+3x)(x^{2}+5−3x) = (x^{2}+5)^{2}−(3x)^{2 }= m^{2}−n^{2}

∴ m = x^{2}+5

QUESTION: 18

Divide 6p^{5}+18p^{4}−3p^{2} by 3p^{2}.

Solution:

QUESTION: 19

Find the factors of b^{2}−7b+12.

Solution:

b^{2 }- 7b+12=b^{2}-3b-4b+12=b(b-3)-4(b-3)=(b-3)(b-4)

QUESTION: 20

Find the factors of6 mn−4n+6−9m.

Solution:

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