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Class 9 Math Algebric Manipulation 3. Find the H.C.F. of the following by division method.(iii) 2 x^{5}-4 x^{4}-6 x x^{5}+x^{4}-3 x^{3}-3 x^{2}


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3. Find the H.C.F. of the following by division method.(iii) 2 x^{5}-4 x^{4}-6 x x^{5}+x^{4}-3 x^{3}-3 x^{2}

6. For what value of k is (x+4) the H.C.F. of x^{2}+x-(2 k+2) and 2 x^{2}+k x-12 ?
6. For what value of  k  is  (x+4)  the H.C.F. of  x^{2}+x-(2 k+2)  and  2 x^{2}+k x-12  ?

6. For what value of k is (x+4) the H.C.F. of x^{2}+x-(2 k+2) and 2 x^{2}+k x-12 ?

2. Fill in the blanks.(iii) HCF of y^{2}-5 y and y+6 and y-2 is
2. Fill in the blanks.(iii) HCF of  y^{2}-5 y  and  y+6  and  y-2  is

2. Fill in the blanks.(iii) HCF of y^{2}-5 y and y+6 and y-2 is

2. Find the square root of the following algebraic expressions by division method.(viii) x^{6}+\frac{1}{x^{6}}-4\left(x^{3}+\frac{1}{x^{3}}\right)+6 x \neq 0
2. Find the square root of the following algebraic expressions by division method.(viii)  x^{6}+\frac{1}{x^{6}}-4\left(x^{3}+\frac{1}{x^{3}}\right)+6 x \neq 0

2. Find the square root of the following algebraic expressions by division method.(viii) x^{6}+\frac{1}{x^{6}}-4\left(x^{3}+\frac{1}{x^{3}}\right)+6 x \neq 0

Example 01 Simplify: \frac{x^{2}-x-6}{2 x^{2}-5 x-3}+\frac{1}{4 x^{2}-1}
Example 01 Simplify:  \frac{x^{2}-x-6}{2 x^{2}-5 x-3}+\frac{1}{4 x^{2}-1}

Example 01 Simplify: \frac{x^{2}-x-6}{2 x^{2}-5 x-3}+\frac{1}{4 x^{2}-1}

Simplify the following\[\text { (x) } 2\left(\frac{x^{2}+7 x+12}{x^{2}-16}+\frac{x^{2}+x-2}{x^{2}+4 x+4}\right) \times \frac{x^{2}-2 x-8}{2 x+8 x^{2}+4} .\]
Simplify the following\[\text { (x) } 2\left(\frac{x^{2}+7 x+12}{x^{2}-16}+\frac{x^{2}+x-2}{x^{2}+4 x+4}\right) \times \frac{x^{2}-2 x-8}{2 x+8 x^{2}+4} .\]

Simplify the following\[\text { (x) } 2\left(\frac{x^{2}+7 x+12}{x^{2}-16}+\frac{x^{2}+x-2}{x^{2}+4 x+4}\right) \times \frac{x^{2}-2 x-8}{2 x+8 x^{2}+4} .\]

9. Let p(x)=10\left(x^{2}-9\right)\left(x^{2}-3 x+2\right) and q(x)=10 x(x+3)(x-1)^{2} . If the H.C.F. of p(x) q(x) is 10(x+3)(x-1) find their L.C.M.
9. Let  p(x)=10\left(x^{2}-9\right)\left(x^{2}-3 x+2\right)  and  q(x)=10 x(x+3)(x-1)^{2} . If the H.C.F. of  p(x) q(x)  is  10(x+3)(x-1)  find their L.C.M.

9. Let p(x)=10\left(x^{2}-9\right)\left(x^{2}-3 x+2\right) and q(x)=10 x(x+3)(x-1)^{2} . If the H.C.F. of p(x) q(x) is 10(x+3)(x-1) find their L.C.M.

Example 04 Simplify: \frac{p^{2}-q^{2}}{r^{2}+2 r s+s^{2}} \div \frac{2(p+q)}{3(r+s) s}
Example 04 Simplify:   \frac{p^{2}-q^{2}}{r^{2}+2 r s+s^{2}} \div \frac{2(p+q)}{3(r+s) s}

Example 04 Simplify: \frac{p^{2}-q^{2}}{r^{2}+2 r s+s^{2}} \div \frac{2(p+q)}{3(r+s) s}

Tick (\boldsymbol{\nu}) the correct answers(vi) Simplified form of \frac{y}{25 x^{2}-y^{2}}-\frac{1}{5 x-y} is:(a) \frac{5 x}{25 x^{2}-y^{2}} (b) \frac{5 x}{5 x^{2}-y} (c) \frac{-5 x}{5 x+y} (d) \frac{-5 x}{25 x^{2}-y^{2}}
Tick  (\boldsymbol{\nu})  the correct answers(vi) Simplified form of  \frac{y}{25 x^{2}-y^{2}}-\frac{1}{5 x-y}  is:(a)  \frac{5 x}{25 x^{2}-y^{2}} (b)  \frac{5 x}{5 x^{2}-y} (c)  \frac{-5 x}{5 x+y} (d)  \frac{-5 x}{25 x^{2}-y^{2}}

Tick (\boldsymbol{\nu}) the correct answers(vi) Simplified form of \frac{y}{25 x^{2}-y^{2}}-\frac{1}{5 x-y} is:(a) \frac{5 x}{25 x^{2}-y^{2}} (b) \frac{5 x}{5 x^{2}-y} (c) \frac{-5 x}{5 x+y} (d) \frac{-5 x}{25 x^{2}-y^{2}}

3. Find the LCM of the following expressions by factorization method:(ii) 24 p^{2} q^{3} r^{4} 104 p^{5} q^{4} r^{5} and 300 p^{3} q r^{8}
3. Find the LCM of the following expressions by factorization method:(ii)  24 p^{2} q^{3} r^{4} 104 p^{5} q^{4} r^{5}  and  300 p^{3} q r^{8}

3. Find the LCM of the following expressions by factorization method:(ii) 24 p^{2} q^{3} r^{4} 104 p^{5} q^{4} r^{5} and 300 p^{3} q r^{8}

4. Find the L.C.M. of the following by factrization. 12 x^{2}-756 x^{2}-13 x-54 x^{2}-20 x+25
4. Find the L.C.M. of the following by factrization.  12 x^{2}-756 x^{2}-13 x-54 x^{2}-20 x+25

4. Find the L.C.M. of the following by factrization. 12 x^{2}-756 x^{2}-13 x-54 x^{2}-20 x+25

7. Find square root by using factorization.\[\left(x^{2}+\frac{1}{x^{2}}\right)+10\left(x+\frac{1}{x}\right)+27 (x \neq 0)\]
7. Find square root by using factorization.\[\left(x^{2}+\frac{1}{x^{2}}\right)+10\left(x+\frac{1}{x}\right)+27 (x \neq 0)\]

7. Find square root by using factorization.\[\left(x^{2}+\frac{1}{x^{2}}\right)+10\left(x+\frac{1}{x}\right)+27 (x \neq 0)\]

2. Find the square root of the following algebraic expressions by division method.(i) x^{4}+2 x^{3}+3 x^{2}+2 x+1
2. Find the square root of the following algebraic expressions by division method.(i)  x^{4}+2 x^{3}+3 x^{2}+2 x+1

2. Find the square root of the following algebraic expressions by division method.(i) x^{4}+2 x^{3}+3 x^{2}+2 x+1

Simplify each of the following as a rational expression.4. \frac{(x+2)(x+3)}{x^{2}-9}+\frac{(x+2)\left(2 x^{2}-32\right)}{(x-4)\left(x^{2}-x-6\right)}
Simplify each of the following as a rational expression.4.  \frac{(x+2)(x+3)}{x^{2}-9}+\frac{(x+2)\left(2 x^{2}-32\right)}{(x-4)\left(x^{2}-x-6\right)}

Simplify each of the following as a rational expression.4. \frac{(x+2)(x+3)}{x^{2}-9}+\frac{(x+2)\left(2 x^{2}-32\right)}{(x-4)\left(x^{2}-x-6\right)}

5. Find the L.C.M. of the following expressions by factorization.(iii) 2\left(x^{4}-y^{4}\right) 3\left(x^{3}+2 x^{2} y-x y^{2}-2 y^{3}\right)
5. Find the L.C.M. of the following expressions by factorization.(iii)  2\left(x^{4}-y^{4}\right) 3\left(x^{3}+2 x^{2} y-x y^{2}-2 y^{3}\right)

5. Find the L.C.M. of the following expressions by factorization.(iii) 2\left(x^{4}-y^{4}\right) 3\left(x^{3}+2 x^{2} y-x y^{2}-2 y^{3}\right)

2. Find the HCF of the following expressions by division method:(iv) x^{2}+x y-2 y^{2} x^{2}+3 y+2 y^{2} and x^{3}+3 x^{2} y+2 x y^{2}+2 y^{3}
2. Find the HCF of the following expressions by division method:(iv)  x^{2}+x y-2 y^{2} x^{2}+3 y+2 y^{2}  and  x^{3}+3 x^{2} y+2 x y^{2}+2 y^{3}

2. Find the HCF of the following expressions by division method:(iv) x^{2}+x y-2 y^{2} x^{2}+3 y+2 y^{2} and x^{3}+3 x^{2} y+2 x y^{2}+2 y^{3}

3. Tick (\boldsymbol{\nu}) the correct answers(iv) LCM of (x-y)^{4} and (x-y)^{3} is:(a) (x-y) (b) (x-y)^{3} (c) (x-y)^{4} (d) (x-y)^{7}
3. Tick  (\boldsymbol{\nu})  the correct answers(iv) LCM of  (x-y)^{4}  and  (x-y)^{3}  is:(a)  (x-y) (b)  (x-y)^{3} (c)  (x-y)^{4} (d)  (x-y)^{7}

3. Tick (\boldsymbol{\nu}) the correct answers(iv) LCM of (x-y)^{4} and (x-y)^{3} is:(a) (x-y) (b) (x-y)^{3} (c) (x-y)^{4} (d) (x-y)^{7}

(vii) H.C.F. of x^{2}+3 x+2 x^{2}+4 x+3 and x^{2}+5 x+4 is(a) x+1 (b) (x+1)(x+2) (c) x+3 (d) (x+4)(x+1)
(vii) H.C.F. of  x^{2}+3 x+2 x^{2}+4 x+3  and  x^{2}+5 x+4  is(a)  x+1 (b)  (x+1)(x+2) (c)  x+3 (d)  (x+4)(x+1)

(vii) H.C.F. of x^{2}+3 x+2 x^{2}+4 x+3 and x^{2}+5 x+4 is(a) x+1 (b) (x+1)(x+2) (c) x+3 (d) (x+4)(x+1)

Example 01 Find the HCF by division method of the following polynomials: 2 x^{3}+7 x^{2}+4 x-4 and 2 x^{3}+9 x^{2}+11 x+2 .
Example 01 Find the HCF by division method of the following polynomials:  2 x^{3}+7 x^{2}+4 x-4  and  2 x^{3}+9 x^{2}+11 x+2 .

Example 01 Find the HCF by division method of the following polynomials: 2 x^{3}+7 x^{2}+4 x-4 and 2 x^{3}+9 x^{2}+11 x+2 .

(xi) Simplify \frac{a}{9 a^{2}-b^{2}}+\frac{1}{3 a-b}= (a) \frac{4 a}{9 a^{2}-b^{2}} (b) \frac{4 a-b}{9 a^{2}-b^{2}} (c) \frac{4 a+b}{9 a^{2}-b^{2}} (d) \frac{b}{9 a^{2}-b^{2}}
(xi) Simplify  \frac{a}{9 a^{2}-b^{2}}+\frac{1}{3 a-b}= (a)  \frac{4 a}{9 a^{2}-b^{2}} (b)  \frac{4 a-b}{9 a^{2}-b^{2}} (c)  \frac{4 a+b}{9 a^{2}-b^{2}} (d)  \frac{b}{9 a^{2}-b^{2}}
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(xi) Simplify \frac{a}{9 a^{2}-b^{2}}+\frac{1}{3 a-b}= (a) \frac{4 a}{9 a^{2}-b^{2}} (b) \frac{4 a-b}{9 a^{2}-b^{2}} (c) \frac{4 a+b}{9 a^{2}-b^{2}} (d) \frac{b}{9 a^{2}-b^{2}}

5. Find the L.C.M. of the following expressions by factorization.(i) x^{2}-25 x+100 and x^{2}-x-20
5. Find the L.C.M. of the following expressions by factorization.(i)  x^{2}-25 x+100  and  x^{2}-x-20
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5. Find the L.C.M. of the following expressions by factorization.(i) x^{2}-25 x+100 and x^{2}-x-20

3. Find the H.C.F. of the following by division method.(iii) 2 x^{5}-4 x^{4}-6 x x^{5}+x^{4}-3 x^{3}-3 x^{2}
3. Find the H.C.F. of the following by division method.(iii)  2 x^{5}-4 x^{4}-6 x  x^{5}+x^{4}-3 x^{3}-3 x^{2}
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3. Find the H.C.F. of the following by division method.(iii) 2 x^{5}-4 x^{4}-6 x x^{5}+x^{4}-3 x^{3}-3 x^{2}

Example 2Find the square root of the expression\[4 \frac{x^{2}}{y^{2}}+8 \frac{x}{y}+16+12 \frac{y}{x}+9 \frac{y^{2}}{x^{2}}\]
Example 2Find the square root of the expression\[4 \frac{x^{2}}{y^{2}}+8 \frac{x}{y}+16+12 \frac{y}{x}+9 \frac{y^{2}}{x^{2}}\]
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Example 2Find the square root of the expression\[4 \frac{x^{2}}{y^{2}}+8 \frac{x}{y}+16+12 \frac{y}{x}+9 \frac{y^{2}}{x^{2}}\]

2. Fill in the blanks.(i) There are polynomials. methods to be used to find the HCF of
2. Fill in the blanks.(i) There are polynomials. methods to be used to find the HCF of
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2. Fill in the blanks.(i) There are polynomials. methods to be used to find the HCF of

Perform the indicated operations and simplify to the lowest form.9. \frac{x^{2}+x-6}{x^{2}-x-6} \times \frac{x^{2}-4}{x^{2}-9}
Perform the indicated operations and simplify to the lowest form.9.  \frac{x^{2}+x-6}{x^{2}-x-6} \times \frac{x^{2}-4}{x^{2}-9}
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Perform the indicated operations and simplify to the lowest form.9. \frac{x^{2}+x-6}{x^{2}-x-6} \times \frac{x^{2}-4}{x^{2}-9}

3. Tick (\boldsymbol{)}) the correct answers(v) Simplified form of \frac{1}{x+y}+\frac{y}{x^{2}-y^{2}} is:(a) \frac{y+1}{x^{2}-y^{2}} (b) \frac{x}{x^{2}-y^{2}} (c) \frac{y}{x^{2}-y^{2}} (d) \frac{x+y}{x^{2}-y^{2}}
3. Tick  (\boldsymbol{)})  the correct answers(v) Simplified form of  \frac{1}{x+y}+\frac{y}{x^{2}-y^{2}}  is:(a)  \frac{y+1}{x^{2}-y^{2}} (b)  \frac{x}{x^{2}-y^{2}} (c)  \frac{y}{x^{2}-y^{2}} (d)  \frac{x+y}{x^{2}-y^{2}}
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3. Tick (\boldsymbol{)}) the correct answers(v) Simplified form of \frac{1}{x+y}+\frac{y}{x^{2}-y^{2}} is:(a) \frac{y+1}{x^{2}-y^{2}} (b) \frac{x}{x^{2}-y^{2}} (c) \frac{y}{x^{2}-y^{2}} (d) \frac{x+y}{x^{2}-y^{2}}

Simplify the following(viii) \left(\frac{1}{x^{2}-9}\right) \div\left(\frac{1}{x+3}\right)-\frac{3}{x-2}
Simplify the following(viii)  \left(\frac{1}{x^{2}-9}\right) \div\left(\frac{1}{x+3}\right)-\frac{3}{x-2}
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Simplify the following(viii) \left(\frac{1}{x^{2}-9}\right) \div\left(\frac{1}{x+3}\right)-\frac{3}{x-2}

3. Find the LCM of the following expressions by factorization method:(iii) 21 x^{2}-14 x and 3 x^{2}-5 x+2
3. Find the LCM of the following expressions by factorization method:(iii)  21 x^{2}-14 x  and  3 x^{2}-5 x+2
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3. Find the LCM of the following expressions by factorization method:(iii) 21 x^{2}-14 x and 3 x^{2}-5 x+2

(iv) H.C.F. a^{3}+b^{3} and a^{2}-a b+b^{2} is(a) a+b (b) a^{2}-a b+b^{2} (c) (a-b)^{2} (d) a^{2}+b^{2}
(iv) H.C.F.  a^{3}+b^{3}  and  a^{2}-a b+b^{2}  is(a)  a+b (b)  a^{2}-a b+b^{2} (c)  (a-b)^{2} (d)  a^{2}+b^{2}
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(iv) H.C.F. a^{3}+b^{3} and a^{2}-a b+b^{2} is(a) a+b (b) a^{2}-a b+b^{2} (c) (a-b)^{2} (d) a^{2}+b^{2}

1. Find the H.C.F. of the following expressions.(i) 39 x^{7} y^{3} z and 91 x^{5} y^{6} z^{7}
1. Find the H.C.F. of the following expressions.(i)  39 x^{7} y^{3} z  and  91 x^{5} y^{6} z^{7}
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1. Find the H.C.F. of the following expressions.(i) 39 x^{7} y^{3} z and 91 x^{5} y^{6} z^{7}

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