Topics

The value of \( \Large \frac{ \left(x-y\right)^{3}+ \left(y-z\right)^{3}+ \left(z-x\right)^{3} }{ \left(x^{2}-y^{2}\right)^{3}+ \left(y^{2}-z^{2}\right)^{3} \left(z^{2}-x^{2}\right)^{3} } \)


A) 1

B) \( \Large \left[ 2 \left(x+y+z\right) \right]^{-1} \)

C) \( \Large \left[ \left(x+y\right) \left(y+z\right) \left(z+x\right) \right]^{-1} \)

D) 0

Correct Answer:
C) \( \Large \left[ \left(x+y\right) \left(y+z\right) \left(z+x\right) \right]^{-1} \)

Description for Correct answer:
We know, \( \Large a+b+c=0 \)

Therefore, \( \Large a^{3+}b^{3}+c^{3} = 3abc \)

As, \( \Large \left(x^{2}-y^{2}\right)+ \left(y^{2}-z^{2}\right)+ \left(z^{2}-x^{2}\right) = 0 \)

Therefore, \( \Large \left(x^{2}-y^{2}\right)^{3}+ \left(y^{2}-z^{2}\right)^{3}+ \left(z^{2}-x^{2}\right)^{3} \)

= \( \Large 3 \left(x^{2}-y^{2}\right) \left(y^{2}-z^{2}\right) \left(z^{2}-x^{2}\right) \)

= \( \Large 3 \left(x-y\right) \left(x+y\right) \left(y-z\right) \left(y+z\right) \left(z+x\right) \left(z-x\right) \)

=\( \Large 3 \left(x-y\right) \left(y-z\right) \left(z-x\right) \left(x+y\right) \left(y+z\right) \left(z+x\right) \)

Also, \( \Large \left(x-y\right)+ \left(y-z\right)+ \left(z-x\right) = 0 \)

and \( \Large \left(x-y\right)^{3}+ \left(y-z\right)^{3}+ \left(z-x\right)^{3} = 3 \left(x-y\right) \left(y-z\right) \left(z-x\right) \)

Therfore, Given expression

= \( \Large \frac{3 \left(x-y\right) \left(y-z\right) \left(z-x\right) }{3 \left(x-y\right) \left(y-z\right) \left(z-x\right) \left(x+y\right) \left(y+z\right) \left(z+x\right) } \)

= \( \Large \left[ \left(x+y\right) \left(y+z\right) \left(z+x\right) \right]^{-1} \)

Part of solved Polynomials questions and answers : >> Elementary Mathematics >> Polynomials




Comments
No comments available




Similar Questions
1). If remainder of the polynomial \( \Large x^{3}+x^{2}+x+a \) when divided by \( \Large \left(x-2\right) \) is zero, then value of a is
A). -10
B). -12
C). -14
D). -16
-- View Answer
2). If \( \Large \left(x+1\right) \) is a factor of \( \Large x^{4}+9x^{3}+7x^{2}+9ax+5a^{2} \), then
A). \( \Large a = \sqrt{137} \)
B). \( \Large 5a^{2}-9a-1=0 \)
C). \( \Large 5a^{2}-9a-17=0 \)
D). \( \Large a = \sqrt{131} \)
-- View Answer
3). H.C.F. of the polynomials \( \Large x^{2}+3x+2 \) and \( \Large x^{3}+3x^{2}+3x+1 \) using Euclidean algorithm is
A). \( \Large x+4 \)
B). \( \Large 3x+2 \)
C). \( \Large 2x+1 \)
D). \( \Large x+1 \)
-- View Answer
4). Let I, m and n be real dividing such that \( \Large l + m \ne n \). What is the quotient on dividing \( \Large l^{3}-m^{3}+n^{3}+3lmn \) by \( \Large \left(l-m+n\right) \)?
A). \( \Large l^{2}+m^{2}+n^{2}-lm-mn-ln \)
B). \( \Large l^{2}+m^{2}+n^{2}+lm+mn-ln \)
C). \( \Large l^{2}+m^{2}+n^{2}+lm+mn+ln \)
D). \( \Large l^{2}-m^{2}+n^{2}-lm-mn+ln \)
-- View Answer
5). What is the. remainder when \( \Large x^{4}+1 \) is divided \( \Large x-2 \)?
A). 17
B). 15
C). 7
D). 1
-- View Answer


6). If \( \Large x+y+z=0 \) and \( \Large xyz \ne 0 \), then what is the value of \( \Large \frac{1}{x^{2}+y^{2}-z^{2}}+\frac{1}{y^{2}+z^{2}-x^{2}}+\frac{1}{z^{2}+x^{2}-y^{2}} \)
A). 0
B). 1
C). -1
D). \( \Large \frac{1}{2} \)
-- View Answer
7). For what value of p, is the coefficient of \( \Large x^{2} \) in the product \( \Large \left(2x-1\right) \left(x-k\right) \left(px+1\right) \) equal to 0 and the constant term equal to 2?
A). 2
B). \( \Large \frac{2}{5} \)
C). \( \Large \frac{5}{2} \)
D). 5
-- View Answer
8). If m and n are natural numbers such that \( \Large 2^{m}-2^{n}=960 \), then what is the value of m?
A). 10
B). 12
C). 16
D). Cannot be determined
-- View Answer
9). When \( \Large x^{3}+2x^{2}+4x+b \) is divided by \( \Large x+1 \), the quotient is \( \Large x^{2}+ax+3 \) and remainder is \( \Large -3 +2b \). What are the values of a and b respectively?
A). 1, 0
B). 0,1
C). 1, 1
D). -1, -1
-- View Answer
10). If \( \Large y+\frac{1}{z}=1\ and\ x+\frac{1}{y}=1 \), then what is the value of xyz?
A). 1
B). -1
C). 0
D). \( \Large \frac{1}{2} \)
-- View Answer
Home About Us Terms and Conditions Privacy Policy Contact us