Algebraic Expression exercises – 2

shariful Gurukul

13 years ago

Today is our topic of discussion Algebraic Expression exercises – 2 .

Algebraic Expression exercises – 2

1. Which one of the following expression is symmetric?

1) a+b+c

2) xy-yz + zx

3) x2 y2+z2

4) 2a²-5bc – c²

2. if P(x, y, z) = x³ + y³ + 2³ − 3xyz then

(i) P(x, y, z) cyclic
(ii) P(x, y, z) symmetric
(iii) P(1, -2, 1) = 0

which one of the following is correct?
1) i, ii
2) i, iii
3) ii, iii
4) i, ii, iii

If one of the factor of 23+ pr² – x – 7 polynomial is x + 7 then answer the
following 3 and 4.

3. What is the value of p?
1) -7
2) 7
3) 54 7
4) 477

4. What is the product of the other factors of the polynomial?

1) (x-1)(x-1)

2) (x+1)(x-2)

3) (x-1)(x+3)

4) (x+1)(x-1)

5. If a factor of polynomial x4 – 5×3+7x²-a is x-2 then, show that, a = = 4

6. Suppose, P(x) = ax5 + bx² + cx³ + cx² + bx+ a where a, b, c are constant and a0. Show that, if x – r is a factor of P(r), then another factor of P(x) will be (rx-1)

7. Resolve into factors:

1) x²+7x³ + 17x² + 17x+6

2) 4a4+12a3+7a2-3a-2

3) x³+2x²+2x+1

4) x(y²+z²)+y(z² + x²) + z(x² + y²) + 3xyz

5) (x+1)2(y-z) + (y + 1)²(z − x) + (z+1)² (x − y)

6) b²c²(b² – c²)+c²a² (c² — a²) + a²b² (a² — b²)

7) 15×2+2xy-24y2-x+24y6

8) 15x224y²-6×2+2xy – xz + 24yz

8. if + + 1 3 = a3 b3 C3 abc’ show that, bc + ca+ab = 0 or, a = b = c

9. if x = b+ca, y = c +ab, and z = a+b-c show that, x3+y3+23-3xyz = 4(a³ + b³ + c³-3abc)

10. Simplify:

11. Express as partial fraction:

12. The polynomial of x, y, z, F(x, y, z) = x³ + y³ + z³ — 3xyz

1) Show that, F(x, y, z) is a cyclic expression.
2) Factorize F(x, y, z) and if F(x, y, z) = 0, (x + y + z) 0 show that, x² + y²+2² = xy + yz + zx
3) if x = (b+ca), y = (c+a – b) and z = (a+b-c) show that, F(a, b, c): F(x, y, z) = 1:4

13. P(a, b, c) = (a+b+c) (ab+be+ca) and Qa-3+b-3+c-3-3a-¹b-¹c-1 1)

1. Express with proper reason whether P(a, b, c) cyclic or symmetric.

2) if Q0 prove that, a = b c or ab+be+ca = 0
3) if P(a, b, c) abc show that, =

14. P(x) = 18x³ + bx2-x-2 and Q(x) = 4×4 + 12x³ + 7x² – 3x − 2
1) Determine the degree of the qoutient Q(x) P(x)
2) if 3x+2 is a factor of P(x), find the value of b.
3) Express 8×2 – 2 Q(x) as partial fraction.

15. Two polynomials of variable x are P(x) = 7x² – 3x + 4×4 – a + 12x³ and Q(x) = 6×3 + x² – 9x+26

1) Expressing P(x) ideally determine leading co-efficient.
2) A factor of P(x) is (x+2) then find the value of a.
3) Show that there is a common factor of P(x) and Q(x).

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