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Factorize : p^3(q-r)^3+q^3(r-p)^3+r^3(p-...

Factorize : `p^3(q-r)^3+q^3(r-p)^3+r^3(p-q)^3`

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To factorize the expression \( p^3(q - r)^3 + q^3(r - p)^3 + r^3(p - q)^3 \), we can use the identity for the sum of cubes. Here are the steps: ### Step 1: Identify the components Let: - \( a = p(q - r) \) - \( b = q(r - p) \) - \( c = r(p - q) \) ...
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Prove that p x^(q-r)+q x^(r-p)+r x^(p-q)> p+q+r ,where p, q, r are distinct and x!=1.

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If q is the mean proportional between p and r , show that : pqr (p + q + r)^(3) = (pq + qr + pr)^(3) .

If omega is an imaginary cube root of unity, then the value of (p+q)^3+(pomega+qomega^2)^3+(pomega^2+qomega)^3 is: (a) 3(p+q)(p+qomega)(p+qomega^2) (b) 3(p^3+q^3) (c) 3(p^3+q^3)-p q(p+q) (d) 3(p^3+q^3)p q(p+q)

If (x+2) is a common factor of (px^2+qx+r) and (qx^2+px+r) then (a) p=q or p+q+r=0 (b) p=r or p+q+r=0 (c) q=r or p+q+r=0 (d) p=q=-1/2r

If t_n denotes the nth term of an A.P. and t_p= 1/q, t_q= 1/p then which one of the following is necessarily a root of the equation (p+2q-3r)x^2+(q+2r-3p)x+(r+2p-3q)=0 (A) t_p (B) t_q (C) t_(pq) (D) t_(p+q)

The roots of the equation (q-r)x^2+(r-p)x+p-q=0 are (A) (r-p)/(q-r),1 (B) (p-q)/(q-r),1 (C) (q-r)/(p-q),1 (D) (r-p)/(p-q),1

Without expanding the determinant prove that: |(0, p-q, p-r), (q-p,0, q-r),(r-p, r-q, 0)|=0

Let p and q be real numbers such that p!=0,p^3!=q ,and p^3!=-qdot If alpha and beta are nonzero complex numbers satisfying alpha+beta=-p and alpha^3+beta^3=q , then a quadratic equation having alpha//beta and beta//alpha as its roots is A. (p^3+q)x^2-(p^3+2q)x+(p^3+q)=0 B. (p^3+q)x^2-(p^3-2q)x+(p^3+q)=0 C. (p^3+q)x^2-(5p^3-2q)x+(p^3-q)=0 D. (p^3+q)x^2-(5p^3+2q)x+(p^3+q)=0

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RD SHARMA ENGLISH-FACTORIZATION OF ALGEBRAIC EXPRESSIONS-All Questions
  1. Factorize each of the following expressions : (i) 4sqrt(3)x^2+5x-2sqr...

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  2. Factorize : (x^2-4x)(x^2-4x-1)-20

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  3. Factorize : p^3(q-r)^3+q^3(r-p)^3+r^3(p-q)^3

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  4. Factorize : (x-y)^3+(y-z)^3+(z-x)^3

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  5. Factorize : 2sqrt(2)x^3+3sqrt(3)y^3+sqrt(5)(5-3sqrt(6)x y)

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  6. Prove that : a^3+b^3+c^3-3a b c=1/2(a+b+c)"{"a-b")"^2+(b-c)^2+(c-a)^2}

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  7. Factorize : 2sqrt(2)a^3+8b^3-27 c^3+18sqrt(2)a b c

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  8. Resolve a^3-b^3+1+3a b into factors.

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  9. Factorise : (a+b)^3+(b+c)^3+(c+a)^3-3(a+b)(b+c)(c+a)

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  10. Factorize each of the following expressions : x^2+2x y+y^2-a^2+2a...

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  11. Factorize each of the following expressions : 3-12(a-b)^2

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  12. Factorize each of the following expressions : x(x+z)-y(y+z)

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  13. Factorize each of the following expressions : a^2-b^2-a-b

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  14. Factorize each of the following expressions : 25 x^2-10 x+1-36 y^2

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  15. Factorize each of the following expressions : 1-2a b-(a^2+b^2)

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  16. Factorize each of the following expressions: a b(a^2+b^2-c^2)+b c...

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  17. Factorize each of the following expressions: x(x^2+y^2-z^2)+y(-...

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  18. Factorize each of the following expressions by splitting the middle ...

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  19. Factorize each of the following expressions by splitting the middle ...

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  20. Factorize each of the following expressions: (a) 2p(a-b)+3q\ (5a-5b)...

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