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The LCM of the polynomials $18\left({\mathrm{x}}^4-{\mathrm{x}}^3+{\mathrm{x}}^2\right) \mathrm{and} 24\left({\mathrm{x}}^6+{\mathrm{x}}^3\right)$ is ____________.
1.  $72{\mathrm{x}}^2\left(\mathrm{x}+1\right){\left({\mathrm{x}}^2-\mathrm{x}+1\right)}^2$
2.  $72{\mathrm{x}}^3{\left({\mathrm{x}}^2-1\right)}^2\left({\mathrm{x}}^3-1\right)$
3.  $72{\mathrm{x}}^3\left({\mathrm{x}}^3-1\right)$
4.  $72{\mathrm{x}}^3\left({\mathrm{x}}^3+1\right)$

The LCM of the polynomials $\mathrm{f}\left(\mathrm{x}\right)=9\left({\mathrm{x}}^3+{\mathrm{x}}^2+\mathrm{x}\right)$ and $\mathrm{g}\left(\mathrm{x}\right)=3\left({\mathrm{x}}^3+1\right)$ is ________.
1.  $27\mathrm{x}\left(\mathrm{x}+1\right)\left({\mathrm{x}}^2+\mathrm{x}+1\right)$
2.  $9\mathrm{x}\left(\mathrm{x}+1\right)\left({\mathrm{x}}^2+\mathrm{x}+1\right)\left({\mathrm{x}}^2-\mathrm{x}+1\right)$
3.  $9\left(\mathrm{x}+1\right)\left({\mathrm{x}}^2-\mathrm{x}+1\right)\left({\mathrm{x}}^2+\mathrm{x}+1\right)$
4.  $9\mathrm{x}\left(\mathrm{x}+1\right)\left({\mathrm{x}}^2+\mathrm{x}+1\right)$

The LCM of polynomials $14\left({\mathrm{x}}^2-1\right)\left({\mathrm{x}}^2+1\right) \mathrm{and} 18\left({\mathrm{x}}^4-1\right)\left(\mathrm{x}+1\right)$ is ___________.
1.  $126\left(\mathrm{x}+1\right)\left({\mathrm{x}}^2+1\right)\left(\mathrm{x}-1\right)$
2.  $126\left(\mathrm{x}+1\right)\left({\mathrm{x}}^2+1\right)\left({\mathrm{x}}^2-1\right)$
3.  $126{\left(\mathrm{x}+1\right)}^2\left({\mathrm{x}}^2+1\right){\left(\mathrm{x}-1\right)}^2$
4.  $126\left(\mathrm{x}+1\right)\left({\mathrm{x}}^2+1\right){\left(\mathrm{x}-1\right)}^2$

If HCF and LCM of two polynomials P(x) and Q(x) are x(x+p) and 12x² (x$-$p)(x²$-$p²) respectively. If P(x)=4x² (x+p), then Q(x)=_______.
1.  $3\mathrm{x}{\left(\mathrm{x}-\mathrm{p}\right)}^2 \left(\mathrm{x}+\mathrm{p}\right)$
2.  $3\mathrm{x}\left(\mathrm{x}-\mathrm{p}\right) \left(\mathrm{x}+\mathrm{p}\right)$
3.  $3\mathrm{x}\left(\mathrm{x}+\mathrm{p}\right) \left({\mathrm{x}}^2-{\mathrm{p}}^2\right)$
4.  $3\mathrm{x}\left(\mathrm{x}+\mathrm{p}\right) \left({\mathrm{x}}^2+{\mathrm{p}}^2\right)$

The product of HCF and LCM of two polynomials is $\left({\mathrm{x}}^2-1\right)\left({\mathrm{x}}^4-1\right),$ then the product of the polynomials is __________.
1.  $\left({\mathrm{x}}^2-1\right)\left({\mathrm{x}}^2+1\right)$
2.  $\left({\mathrm{x}}^2-1\right){\left({\mathrm{x}}^2+1\right)}^2$
3.  ${\left({\mathrm{x}}^2-1\right)}^2\left({\mathrm{x}}^2+1\right)$
4.  None of these

If (x+4)(x$-$2)(x+1) is the HCF of the polynomials f(x)=(x²+2x$-$8)(x²+4x+a) and g(x) = (x²$-$x$-$2)(x²+3x$-$b), then (a, b) = ________.
1.  (3, $-$4)
2.  ($-$3, $-$4)
3.  ($-$3, 4)
4.  (3, 4)

Find the LCM of ${\mathrm{x}}^3-{\mathrm{x}}^2+\mathrm{x}-1 \mathrm{and} {\mathrm{x}}^3-2{\mathrm{x}}^2+\mathrm{x}-2.$
1.  $\left(\mathrm{x}+1\right)\left(\mathrm{x}-1\right)$
2.  $\mathrm{x}-1$
3.  $\left({\mathrm{x}}^2+1\right)\left(\mathrm{x}-1\right)\left(\mathrm{x}-2\right)$
4.  $\left({\mathrm{x}}^2+1\right)\left(\mathrm{x}+1\right)\left(\mathrm{x}-2\right)$

The HCF of the polynomials p(x) and q(x) is 6x$-$9, then p(x) and q(x) could be ___________.
1.  3, 2x$-$3
2.  12x$-$18, 2
3.  3(2x-3)², 6(2x-3)
4.  3(2x-3), 6(2x+3)

If the HCF of the polynomials f(x) and g(x) is 4x$-$6, then f(x) and g(x) could be ___________.
1.  2, 2x$-$3
2.  8x$-$12, 2
3.  2(2x$-$3)², 4(2x$-$3)
4.  2(2x+3), 4(2x+3)

If the HCF of the polynomials f(x) = (x+3)(3x²$-$7x$-$a) and g(x)=(x$-$3)(2x²+3x+b) is (x+3) (x$-$3), then a+b= __________.
1.  3
2.  $-$15
3.  $-$3
4.  15

Find the HCF of the polynomials $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^3+{\mathrm{x}}^2+\mathrm{x}+1 \mathrm{and} \mathrm{g}\left(\mathrm{x}\right)={\mathrm{x}}^3-{\mathrm{x}}^2+\mathrm{x}-1.$
1.  $\mathrm{x}\left(\mathrm{x}+1\right)$
2.  $\mathrm{x}-1$
3.  ${\mathrm{x}}^2+1$
4.  $\mathrm{x}+1$

If the HCF of ${\mathrm{x}}^3+2{\mathrm{x}}^2-\mathrm{ax} \mathrm{and} 2{\mathrm{x}}^3+5{\mathrm{x}}^2-3\mathrm{x}$ is $\mathrm{x}\left(\mathrm{x}+3\right),$ then a = _________.
1.  3
2.  -3
3.  6
4.  -4

The HCF of the polynomials $70\left({\mathrm{x}}^3-1\right) \mathrm{and} 105\left({\mathrm{x}}^2-1\right)$ is __________.
1.  $15\left(\mathrm{x}-1\right)$
2.  $35\left(\mathrm{x}-1\right)$
3.  $35\left({\mathrm{x}}^2-1\right)\left({\mathrm{x}}^2+\mathrm{x}+1\right)$
4.  $15\left({\mathrm{x}}^2-1\right)$

What should be subtracted from $\frac{7\mathrm{x}}{\left({\mathrm{x}}^2-\mathrm{x}-12\right)}$ to get $\frac{3}{\mathrm{x}+3}?$
1.  $\frac{5}{\mathrm{x}+4}$
2.  $\frac{4}{\mathrm{x}-4}$
3.  $\frac{2}{\mathrm{x}-4}$
4.  $\frac{1}{\mathrm{x}-4}$

Which of the following is/are true?
(A) The sum of two rational expressions is always a rational expression.
(B) The difference of two rational expressions is always a rational expression.
(C) $\frac{\mathrm{p}\left(\mathrm{x}\right)}{\mathrm{q}\left(\mathrm{x}\right)}$ is in its lowest terms of LCM [p(x), q(x)]=1.
(D) Reciprocal of $\frac{-2\mathrm{x}}{{\mathrm{x}}^2-1} \mathrm{is} \frac{{\mathrm{x}}^2-1}{2\mathrm{x}}.$
1.  A, B
2.  A, B, D
3.  A, C
4.  A, B, C

The product of additive inverses of $\frac{{\mathrm{x}}^2-1}{2\mathrm{x}}$ and $\frac{{\mathrm{x}}^2-4}{3-\mathrm{x}}$ is _______.
1.  ${\mathrm{x}}^2+5\mathrm{x}+6$
2.  ${\mathrm{x}}^2+\mathrm{x}-6$
3.  ${\mathrm{x}}^2-\mathrm{x}-6$
4.  ${\mathrm{x}}^2-5\mathrm{x}+6$

What should be added to $\frac{1}{{\mathrm{x}}^2-7\mathrm{x}+12}$ to get $\frac{2}{{\mathrm{x}}^2-6\mathrm{x}+18}?$
1.  $\frac{2}{\left(\mathrm{x}+3\right)\left(\mathrm{x}-2\right)}$
2.  $\frac{4}{\left(\mathrm{x}+3\right)\left(\mathrm{x}+2\right)}$
3.  $\frac{1}{{\mathrm{x}}^2-5\mathrm{x}+6}$
4.  $\frac{-1}{{\mathrm{x}}^2+5\mathrm{x}-6}$

The rational expression $\frac{{\mathrm{x}}^3-3{\mathrm{x}}^2+2\mathrm{x}}{{\mathrm{x}}^2\mathrm{y}-2\mathrm{xy}}$ in lowest terms is ___________.
1.  $\frac{\mathrm{x}+2}{\mathrm{y}}$
2.  $\frac{\mathrm{x}+1}{\mathrm{xy}}$
3.  $\frac{\mathrm{x}-2}{\mathrm{y}}$
4.  $\frac{\mathrm{x}+1}{\mathrm{y}}$

If $\left(\mathrm{x}-4\right)$ is the HCF of $\mathrm{p}\left(\mathrm{x}\right)={\mathrm{x}}^2-\mathrm{nx}-12 \mathrm{and} \mathrm{q}\left(\mathrm{x}\right)={\mathrm{x}}^2-\mathrm{mx}-8,$ then the simplest form of $\frac{\mathrm{p}\left(\mathrm{x}\right)}{\mathrm{q}\left(\mathrm{x}\right)}$ is ________.
1.  $\frac{\mathrm{x}-3}{\mathrm{x}+2}$
2.  $\frac{\mathrm{x}+3}{\mathrm{x}-2}$
3.  $\frac{\mathrm{x}+2}{\mathrm{x}+3}$
4.  $\frac{\mathrm{x}+3}{\mathrm{x}+2}$

What should be added to $\frac{2}{\left({\mathrm{x}}^2+\mathrm{x}-6\right)}$ to get $\frac{-4\mathrm{x}}{\left({\mathrm{x}}^2-4\right)\left({\mathrm{x}}^2-9\right)}?$
1.  $\frac{4}{{\mathrm{x}}^2-\mathrm{x}-6}$
2.  $\frac{-2}{{\mathrm{x}}^2-\mathrm{x}-6}$
3.  $\frac{4\mathrm{x}}{{\mathrm{x}}^2+\mathrm{x}+6}$
4.  $\frac{-3}{{\mathrm{x}}^2+\mathrm{x}+6}$

The HCF of the polynomials $4{\left(\mathrm{x}+8\right)}^2 \left({\mathrm{x}}^2-5\mathrm{x}+6\right) \mathrm{and} 6\left({\mathrm{x}}^2+12\mathrm{x}+32\right)\left({\mathrm{x}}^2-7\mathrm{x}+12\right)$ is __________.
1.  $2\left(\mathrm{x}+8\right)\left(\mathrm{x}-3\right)$
2.  $\left(\mathrm{x}-3\right)\left(\mathrm{x}+8\right)$
3.  $\left(\mathrm{x}-8\right)\left(\mathrm{x}+3\right)$
4.  $2\left(\mathrm{x}-8\right)\left(\mathrm{x}+3\right)$

The LCM of the polynomials $8\left({\mathrm{x}}^2-2\mathrm{x}\right){\left(\mathrm{x}-6\right)}^2 \mathrm{and} 2\mathrm{x}\left({\mathrm{x}}^2-4\right){\left(\mathrm{x}-6\right)}^2$ is __________.
1.  $8\left({\mathrm{x}}^2-4\right){\left({\mathrm{x}}^2-6\mathrm{x}\right)}^2$
2.  $\left({\mathrm{x}}^3-4\mathrm{x}\right){\left(\mathrm{x}-6\right)}^2$
3.  $8\left({\mathrm{x}}^3-4\mathrm{x}\right){\left(\mathrm{x}-6\right)}^2$
4.  $8\mathrm{x}\left({\mathrm{x}}^2+4\right){\left(\mathrm{x}-6\right)}^2$

The HCF of polynomials $\left({\mathrm{x}}^2-2\mathrm{x}+1\right)\left(\mathrm{x}+4\right) \mathrm{and} \left({\mathrm{x}}^2+3\mathrm{x}-4\right)\left(\mathrm{x}+1\right)$ _____________.
1.  $\left(\mathrm{x}+4\right)\left(\mathrm{x}-1\right)$
2.  $\left(\mathrm{x}+1\right)\left(\mathrm{x}+4\right)$
3.  $\left(\mathrm{x}+1\right)\left(\mathrm{x}-4\right)$
4.  $\left({\mathrm{x}}^2-1\right)\left(\mathrm{x}+4\right)$

If the zeroes of the rational expression $\left(3\mathrm{x}+2\mathrm{a}\right)\left(2\mathrm{x}+1\right) \mathrm{are} \frac{-1}{2} \mathrm{and} \frac{\mathrm{b}}{3},$ then the value of a is ___________.
1.  $-2\mathrm{b}$
2.  $\frac{-\mathrm{b}}{2}$
3.  $\frac{-\mathrm{b}}{3}$
4.  None of these

The LCM of the polynomials $\left({\mathrm{x}}^2-8\mathrm{x}+16\right)\left({\mathrm{x}}^2-25\right) \mathrm{and} \left({\mathrm{x}}^2-10\mathrm{x}+25\right)\left({\mathrm{x}}^2-2\mathrm{x}-24\right)$ is ____________.
1.  $\left({\mathrm{x}}^4-41\mathrm{x}+400\right)\left(x-6\right)$
2.  $\left({\mathrm{x}}^4+41\mathrm{x}+400\right)\left(x^{\mathit{2}}-9\mathrm{x}+20\right)$
3.  $\left({\mathrm{x}}^4-41\mathrm{x}+400\right)\left({\mathrm{x}}^2-9\mathrm{x}+20\right)\left(\mathrm{x}-6\right)$
4.  $\left({\mathrm{x}}^4-41\mathrm{x}+400\right)\left({\mathrm{x}}^2-9\mathrm{x}+20\right)\left(\mathrm{x}+6\right)$

If the HCF of the polynomials $\left({\mathrm{x}}^2+8\mathrm{x}+16\right)\left({\mathrm{x}}^2-9\right) \mathrm{and} \left({\mathrm{x}}^2+7\mathrm{x}+12\right) \mathrm{is} {\mathrm{x}}^2+7\mathrm{x}+12$, then their LCM is ____________.
1.  $\left(\mathrm{x}+4\right)\left({\mathrm{x}}^2-9\right)$
2.  ${\left(\mathrm{x}+4\right)}^2 \left({\mathrm{x}}^2-9\right)$
3.  $\left({\mathrm{x}}^2-4\right)\left({\mathrm{x}}^2+9\right)$
4.  ${\left(\mathrm{x}+4\right)}^2 \left(\mathrm{x}+3\right)$

If $\mathrm{h}\left(\mathrm{x}\right)={\mathrm{x}}^2+\mathrm{x} \mathrm{and} \mathrm{g}\left(\mathrm{y}\right)={\mathrm{y}}^3-\mathrm{y},$ then the HCF of h(b)-h(a) and g(b)-g(a) is _________.
1.  $\mathrm{a}+\mathrm{b}$
2.  $\mathrm{b}-\mathrm{a}$
3.  ${\mathrm{b}}^2+{\mathrm{a}}^2$
4.  ${\mathrm{b}}^2+\mathrm{ab}+{\mathrm{a}}^2$

If (y)=y³ and g(z)=z⁴, then HCF of h(b)$-$h(a) and g(b)$-$g(a) is ___________.
1.  b $-$ a
2.  b² $-$ a²
3.  b³ $-$ a³
4.  b²+ab+a²

If the HCF of the polynomials (x+4)(2x²+5x+a) and (x+3)(x²+7x+b) is (x²+7x+12) then 6a+b is ________
1.  -6
2.  5
3.  6
4.  -5

The HCF and LCM of the polynomials p(x) and q(x) are 5(x-2)(x+9) and 10(x²+16x+63)(x-2)². If p(x) is 10(x+9)(x²+5x-14), then q(x) is ________.
1.  5(x+9)(x-2)
2.  10(x-2)² (x+7)
3.  10(x+9)(x-2)
4.  5(x-2)² (x+9)

If the zeroes of the rational expression (ax+b)(3x+2) are $\frac{-2}{3} \mathrm{and} \frac{1}{2},$ then a+b=____________.
1.  -1
2.  0
3.  -b
4.  -a

Simplify:
$\frac{{\mathrm{x}}^2-{\left(\mathrm{y}-2\mathrm{z}\right)}^2}{\mathrm{x}-\mathrm{y}+2\mathrm{z}}+\frac{{\mathrm{y}}^2-{\left(2\mathrm{x}-\mathrm{z}\right)}^2}{\mathrm{y}+2\mathrm{x}-\mathrm{z}}+\frac{{\mathrm{z}}^2-{\left(\mathrm{x}-2\mathrm{y}\right)}^2}{\mathrm{z}-\mathrm{x}+2\mathrm{y}}.$
1.  0
2.  1
3.  x+y+z
4.  None of these

If the HCF of the polynomials x² + px +q and x² + ax + b is x + l, then their LCM is ____________.
1.  (x + a - l) ( x + l - p)
2.  (x - (l + a)) (x + l - p) (x + l)
3.  (x + a - l) (x + p - l) (x + l)
4.  (x - l + a) (x - p + l) (x + l)

The expression $\frac{1}{1-\mathrm{x}}-\frac{1}{1+x}-\frac{x^3}{1-x}+\frac{x^2}{1+x}$ is lowest terms is _________.
1.  $2{\mathrm{x}}^3+1$
2.  ${\mathrm{x}}^2+2$
3.  ${\mathrm{x}}^2+2\mathrm{x}$
4.  ${\mathrm{x}}^2-2\mathrm{x}$

Simplify:
$\frac{{\mathrm{a}}^2-{\left(\mathrm{b}-\mathrm{c}\right)}^2}{{\left(\mathrm{a}+\mathrm{c}\right)}^2-{\mathrm{b}}^2}+\frac{{\mathrm{b}}^2-{\left(\mathrm{a}-\mathrm{c}\right)}^2}{{\left(\mathrm{a}+\mathrm{b}\right)}^2-{\mathrm{c}}^2}+\frac{{\mathrm{c}}^2-{\left(\mathrm{a}-\mathrm{b}\right)}^2}{{\left(\mathrm{b}+\mathrm{c}\right)}^2-{\mathrm{a}}^2}.$
1.  0
2.  1
3.  $\mathrm{a}+\mathrm{b}+\mathrm{c}$
4.  $\frac{1}{\mathrm{a}+\mathrm{b}+\mathrm{c}}$

Simplify:
$\frac{\mathrm{x}+1}{\mathrm{x}-1}+\frac{\mathrm{x}-1}{\mathrm{x}+1}-\frac{2{\mathrm{x}}^2-2}{{\mathrm{x}}^2+1}.$
1.  $\frac{4{\mathrm{x}}^4+2}{{\mathrm{x}}^4-1}$
2.  $\frac{4{\mathrm{x}}^2}{{\mathrm{x}}^4-1}$
3.  $\frac{8{\mathrm{x}}^2}{{\mathrm{x}}^4-1}$
4.  1

If the LCM of the polynomials ${\mathrm{x}}^9+{\mathrm{x}}^6+{\mathrm{x}}^3+1$ and ${\mathrm{x}}^6-1$ is ${\mathrm{x}}^{12}-1,$ then their HCF is _________.
1.  ${\mathrm{x}}^3+1$
2.  ${\mathrm{x}}^6+1$
3.  ${\mathrm{x}}^3-1$
4.  ${\mathrm{x}}^6-1$

If x²+x-1 is a factor of x⁴+px³+qx²-1, then the values of p and q can be
1.  2, 1
2.  1, -2
3.  -1, -2
4.  -2, -1

The HCF of two polynomials p(x) and q(x) using long division method was found to be x+5, if their first three quotients obtained are x, 2x+5, and x+3 respectively. Find p(x) and q(x). (The degree of p(x) > the degree of q(x))
1.  p(x)=2x⁴+21x³+72x²+88x+15
     q(x)=2x³+21x²+71x+80
2.  p(x)=2x⁴-21x³-72x²-88x+15
     q(x)=2x³+21x²-71x+80
3.  p(x)=2x⁴+21x³+88x+15
     q(x)=2x³+71x+80
4.  p(x)=2x⁴-21x²-72x²+80x+15
     q(x)=2x³-21x²+71x+80

If the HCF of the polynomials x³+px+q and x³+rx²+lx+x is x²+ax+b, then their LCM is _________. (r$\ne$0)
1.  (x² + ax + b) (x + a) (x + a - r)
2.  (x² + ax + b) (x - a) (x - a + r)
3.  (x² + ax + b) (x - a) (x - a - r)
4.  (x² - ax + b) (x - a) (x - a + r)

If the HCF of the polynomials (x-3)(3x²+10x+b) and (3x-2)(x²-2x+a) is (x-3)(3x-2), then the relation between a and b is _________.
1.  $3\mathrm{a}+8\mathrm{b}=0$
2. $8\mathrm{a}-3\mathrm{b}=0$
3.  $8\mathrm{a}+3\mathrm{b}=0$
4.  $\mathrm{a}-2\mathrm{b}=0$

The HCF of the polynomials 12(x+2)³ (x²-7x+10) and 18(x²-4)(x²-6x+5) is __________.
1.  (x²+3x+10)(x-2)
2.  6(x²+3x+10)(x+2)
3.  (x²-3x-10)(x-2)
4.  6(x²-3x-10)(x-2)

The HCF of the polynomials (x²-4x+4)(x+3) and (x²+2x-3)(x-2) is ____________.
1.  x+3
2.  x-2
3.  (x+3)(x-2)
4.  (x+3)(x-2)²

The HCF of the polynomials 5(x²-16)(x+8) and 10(x²-64)(x+4) is _________.
1.  x²+12+32
2.  5(x²+12x+32)
3.  x2-12x+32
4.  5(x2-12x+32)

Find the HCF of 6x⁴y and 12xy.
1.  6x²y
2.  6x
3.  6y
4.  6xy

Find the LCM of p⁴a²r³ and q³p⁶r⁵.
1.  p⁴q³r³
2.  p⁴q²r⁵
3.  p⁶q³r⁵
4.  p⁶q²r⁵

The LCM of the polynomials 12(x³+27) and 18(x²-9) is ______________.
1.  6(x+3)
2.  36(x²-9)(x²+3x+9)
3.  36(x+3)² (x²+3x+9)
4.  36(x²-9)(x²-3x+9)

The rational expression $\frac{{\mathrm{x}}^2+2\mathrm{x}+3}{{\mathrm{x}}^4+4{\mathrm{x}}^3+4{\mathrm{x}}^2-9}$ in its lowest terms is __________.
1.  $\frac{1}{{\mathrm{x}}^2+3\mathrm{x}+3}$
2.  $\frac{1}{{\mathrm{x}}^2+2\mathrm{x}-3}$
3.  $\frac{1}{{\mathrm{x}}^2+4\mathrm{x}-3}$
4.  $\frac{1}{{\mathrm{x}}^2+2\mathrm{x}+3}$

The LCM of the polynomials (x+3)² (x-2)(x+1)² and (x+1)³ (x+3)(x²-4) is _________.
1.  (x+1)³ (x+3)(x²-4)
2.  (x+3)² (x+1)³ (x²-4)
3.  (x+3)² (x+1)³ (x+2)
4.  (x+3)² (x+1)² (x-2)

The HCF of two polynomials p(x) and q(x) using long division method was found in two steps to be 3x-2, and the first two quotients obtained are x+2 and 2x+1. Find p(x) and q(x). (The degree of p(x) > the degree of q(x)).
1.  $\mathrm{p}\left(\mathrm{x}\right)=6{\mathrm{x}}^3+11{\mathrm{x}}^2+\mathrm{x}+6, \mathrm{q}\left(\mathrm{x}\right)=6{\mathrm{x}}^2+\mathrm{x}+2$
2.  $\mathrm{p}\left(\mathrm{x}\right)=6{\mathrm{x}}^3+11{\mathrm{x}}^2-\mathrm{x}+6, \mathrm{q}\left(\mathrm{x}\right)=6{\mathrm{x}}^2-\mathrm{x}+2$
3.  $\mathrm{p}\left(\mathrm{x}\right)=6{\mathrm{x}}^3-11{\mathrm{x}}^2+\mathrm{x}-6, \mathrm{q}\left(\mathrm{x}\right)=6{\mathrm{x}}^2-\mathrm{x}-2$
4.  $\mathrm{p}\left(\mathrm{x}\right)=6{\mathrm{x}}^3+11{\mathrm{x}}^2-\mathrm{x}-6, \mathrm{q}\left(\mathrm{x}\right)=6{\mathrm{x}}^2-\mathrm{x}-2$

Simplify: $\frac{\mathrm{x}+2}{\mathrm{x}-2}+\frac{\mathrm{x}-2}{\mathrm{x}+2}-\frac{3{\mathrm{x}}^2-3}{{\mathrm{x}}^2+4}.$
1.  $\frac{-{\mathrm{x}}^4+31{\mathrm{x}}^2+20}{{\mathrm{x}}^4-16}$
2.  $\frac{{\mathrm{x}}^4+31{\mathrm{x}}^2+20}{{\mathrm{x}}^4-16}$
3.  $-\frac{{\mathrm{x}}^4+31{\mathrm{x}}^2-20}{{\mathrm{x}}^4-16}$
4.  $-\frac{{\mathrm{x}}^4-21{\mathrm{x}}^2+20}{{\mathrm{x}}^4-16}$

If $\mathrm{P}=\frac{\mathrm{x}+1}{\mathrm{x}-1} \mathrm{and} \mathrm{Q}=\frac{\mathrm{x}-1}{\mathrm{x}+1},$ then ${\mathrm{P}}^2+{\mathrm{Q}}^2-2\mathrm{PQ}=$_________.
1.  $\frac{16{\mathrm{x}}^2}{{\mathrm{x}}^4-2\mathrm{x}+1}$
2.  $\frac{4{\mathrm{x}}^4+8{\mathrm{x}}^2+4}{{\mathrm{x}}^4-2\mathrm{x}+1}$
3.  $\frac{4{\mathrm{x}}^2}{{\mathrm{x}}^4+2{\mathrm{x}}^2+1}$
4.  $\frac{8{\mathrm{x}}^2}{{\mathrm{x}}^4-2{\mathrm{x}}^2+1}$

Simplify: $\frac{81{\mathrm{x}}^4-16{\mathrm{x}}^2+32\mathrm{x}-16}{9{\mathrm{x}}^2-4\mathrm{x}+4}$.
1.  $9{\mathrm{x}}^2+4\mathrm{x}-4$
2.  $9{\mathrm{x}}^2-4\mathrm{x}-4$
3.  $9{\mathrm{x}}^2-2\mathrm{x}-8$
4.  $9{\mathrm{x}}^2+2\mathrm{x}-8$

The rational expression $\mathrm{A}=\left(\frac{\mathrm{x}+1}{\mathrm{x}-1}-\frac{\mathrm{x}-1}{\mathrm{x}+1}-\frac{4\mathrm{x}}{{\mathrm{x}}^2+1}\right)$ is multiplied with the additice inverse of $\mathrm{B}=\frac{1-{\mathrm{x}}^4}{4\mathrm{x}}$ to get C. Then, C = __________.
1.  $\frac{32{\mathrm{x}}^2}{{\mathrm{x}}^4-1}$
2.  $\frac{2\mathrm{x}}{{\mathrm{x}}^4-1}$
3.  2
4.  1

If the HCF of ${\mathrm{x}}^3+{\mathrm{qx}}^2+\mathrm{px} \mathrm{and} {\mathrm{qx}}^3+11{\mathrm{x}}^2-4\mathrm{x} \mathrm{is} \mathrm{x}\left(\mathrm{x}+4\right),$ then p = ___________.
1.  -4
2.  3
3.  -2
4.  5

If degree of both f(x) and [f(x)+g(x)] is 18, then degree of g(x) can be _______________.
1.  18
2.  9
3.  6
4.  Any one of these

The product of additive inverse and multiplicative inverse of $\frac{\mathrm{x}-2}{{\mathrm{x}}^2-4}$ is ___________.
1.  ${\mathrm{x}}^2+4\mathrm{x}+4$
2.  ${\mathrm{x}}^2-4\mathrm{x}+4$
3.  ${\mathrm{x}}^2-6\mathrm{x}+9$
4.  None of these

What should be multiplied to $\left(2{\mathrm{x}}^2+3\mathrm{x}-4\right)$ to get $4{\mathrm{x}}^4-9{\mathrm{x}}^2+24\mathrm{x}-16?$
1.  $2{\mathrm{x}}^2-3\mathrm{x}-4$
2.  $2{\mathrm{x}}^2+24\mathrm{x}-16$
3.  $2{\mathrm{x}}^2+3\mathrm{x}+4$
4.  $2{\mathrm{x}}^2-3\mathrm{x}+4$

If $\mathrm{f}\left(\mathrm{x}\right)={\left(\mathrm{x}+2\right)}^7{\left(\mathrm{x}+4\right)}^{\mathrm{P}}, \mathrm{g}\left(\mathrm{x}\right)={\left(\mathrm{x}+2\right)}^{\mathrm{q}}{\left(\mathrm{x}+4\right)}^9$ and the LCM of f(x) and g(x) is ${\left(\mathrm{x}+2\right)}^{\mathrm{q}} {\left(\mathrm{x}+4\right)}^9,$ then find the maximum value of $\mathrm{p}-\mathrm{q}.$
1.  4
2.  3
3.  2
4.  1

If $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^2-7\mathrm{x}+12 \mathrm{and} \mathrm{g}\left(\mathrm{x}\right)={\mathrm{x}}^2-8\mathrm{x}+15,$ then find the HCF of f(x) and g(x).
1.  $\mathrm{x}-4$
2.  $\mathrm{x}-3$
3.  $\mathrm{x}-5$
4.  $\mathrm{x}-6$

If $\mathrm{f}\left(\mathrm{x}\right)=\left(\mathrm{x}+2\right)\left({\mathrm{x}}^2+8\mathrm{x}+15\right) \mathrm{and} \mathrm{g}\left(\mathrm{x}\right)=\left(\mathrm{x}+3\right)\left({\mathrm{x}}^2+9\mathrm{x}+20\right), \mathrm{then} \mathrm{find} \mathrm{the} \mathrm{LCM} \mathrm{of} \mathrm{f}\left(\mathrm{x}\right) \mathrm{and} \mathrm{g}\left(\mathrm{x}\right).$
1.  $\left(\mathrm{x}+2\right)\left(\mathrm{x}+3\right)\left(\mathrm{x}+4\right)\left(\mathrm{x}+5\right)$
2.  ${\left(\mathrm{x}+2\right)}^2\left(\mathrm{x}+3\right)\left(\mathrm{x}+5\right)$
3.  $\left(\mathrm{x}+2\right)\left(\mathrm{x}+3\right)\left(\mathrm{x}+5\right)\left(\mathrm{x}+1\right)$
4.  ${\left(\mathrm{x}+2\right)}^2\left(\mathrm{x}+3\right)\left(\mathrm{x}+4\right)$

If $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^2+6\mathrm{x}+\mathrm{a}, \mathrm{g}\left(\mathrm{x}\right)={\mathrm{x}}^2+4\mathrm{x}+\mathrm{b}, \mathrm{h}\left(\mathrm{x}\right)={\mathrm{x}}^2+14\mathrm{x}+\mathrm{c}$ and +14 x+c and the LCM of f(x), g(x) and h(x) is (x+8)(x-2)(x+6), then find a+b+c. (a, b and c are constants).
1.  20
2.  16
3.  32
4.  10

Simplify: $\frac{\left({\mathrm{x}}^2+11\mathrm{x}+28\right)}{\left({\mathrm{x}}^2+13\mathrm{x}+40\right)}+\frac{\left({\mathrm{x}}^2+6\mathrm{x}+8\right)}{\left({\mathrm{x}}^2+11\mathrm{x}+24\right)}.$
1.  $\frac{\left(\mathrm{x}+2\right)\left(\mathrm{x}+7\right)}{\left(\mathrm{x}+3\right)\left(\mathrm{x}+3\right)}$
2.  $\frac{\mathrm{x}+4}{\mathrm{x}+8}$
3.  $\frac{\mathrm{x}+8}{\mathrm{x}+4}$
4.  $\frac{\left(\mathrm{x}+3\right)\left(\mathrm{x}+7\right)}{\left(\mathrm{x}+2\right)\left(\mathrm{x}+5\right)}$

If LCM of f(x) and g(x) is $6{\mathrm{x}}^2+13\mathrm{x}+6,$ then which of the following cannot be the HCF of f(x) and g(x)?
1.  2x+3
2.  3x+1
3.  (2x+3)(3x+2)
4.  3x+2

If the LCM of f(x) and g(x) is ${\mathrm{a}}^6-{\mathrm{b}}^6,$ then their HCF can be ___________.
1.  $\mathrm{a}-\mathrm{b}$
2.  ${\mathrm{a}}^2+\mathrm{ab}+{\mathrm{b}}^2$
3.  ${\mathrm{a}}^2-\mathrm{ab}+{\mathrm{b}}^2$
4.  All of these