8th Class Maths -Polynomials

Topic:- Polynomials

Sample Copy not to be published

SYNOPSIS -1

Introduction :

Algebra is that branch of mathematics which explain the relation among the constants and variables.

Constants and variables :

Generally in Algebra, two types of symbols are used constants and variables (literals)

Constant :

It is a symbol whose value always remains the same, whatever the situation be, it is represented by C (or) K

Ex : 7, 3, 10, -2, e, π

Variable :

It is a symbol whose value changes according the situation

Ex : x, y, z, b, c, y, z, e, c, 9 + +

Algebraic Expression :

An algebraic expression is a collection of terms separated by + and - symbol.

Ex : 7 + 11, 10 - 3, 3x + y, dz - x, x + y

The various parts of an Algebraic expression that are seperated by + (or) - sign are called terms.

Ex : Algebraic expression No. of terms terms

1) -36x 1 -36x

2) ax + y - 5 3 ax, -5y and dz

3) yz - y - 4 7 6, 7/9 and x y

i) Monomial :

Mono means one. An Algebraic expression baving only one term is called a monomial.

Ex : 8, 7, 11, a, xyz etc

ii) Binomial :

“Bi” means two. An algebraic expression having two terms is called a binomial

for Ex : 2, 1, 2, 4, a, ax, by, x, z + etc

iii) Trinomial :

“Tri” means there an algebraic expression having three terms is called a trinomial

POLYNOMIALS

4) Multinomial :

An algebraic expression having two or more terms is called a multinomial

Ex : 3a - 4x + 6y + z

5) Factors and Coefficents :

Each combination of the constants and varilables which form a term is called factor.

Coefficent :

Any factor of a term is called the coefficent of the remaining term

for Ex : i) In ||x, || is coefficent of x

ii) In -5x2y, 5 is coefficent of -x2y, -5 is coefficent of x2y

Note :

0 - = - × - × 13 13 1 13 x

coefficent of x0 is (-13)

6) Definition of polynomial :

A polynomial is an algebraic expression in which each variable involved has power (exponent) a whole number.

Ex : 6 5 11 7 8x x z - + , the power of variables are in 11x6

_____6,

5 - 7x _______5

8z ________8z1_____1

7) Polynomial in one variable :

The algebraic expression like i)12x, ii) 13x - 1 2 1

11 6

3

y y - + etc

Polynomial in two or more variables :

An algebraic expression, whose terms or more variables (literals) such that the exponent of each variable is a whole number is called a polynomial in two or more variables

Ex : 1) 3x2 - 6xy +8y2 is a polynomial in two variables x and y

3 2 y zy xy z + - - 8 15 is a polynomial in three variables x,y and z

Degree of a polynomial

The greatest power (exponent) of the terms of a polynomial is called degree of a polynomial

Ex : 1) 5x3 - 7x8+1 ____ Degree___8

2) 3x _____3x1_____Degree ___1

3) 2m - 7m8 + m13___Degree ___13

8) Zeroes of a polynomial :

If for x = K the value of a polynomial p(x) is ‘0’ i.e p(k) = 0 called zero of the polynomial

a) A zero of polynomial need not be zero

Ex : Let p(x) = 3x +1

∴ Let p(x) = 0 => 3x+1=0=>x = -1/3

∴ -1/3 is zero of the polynomial

Let x4 = 0 => x = 0

∴ ‘0’ is the zero of the polynomial p(x) = x4

(c) Number of zeros of the polynomial is equal to degree of the polynomial

For ex: Let p(x) = x2 -1

Let x2 -1 = 0 => x2 = 1 => x ± 1

∴ Zeroes are -1 & +1

∴ clearly degree is same as the number of zeroes.

(d) A polynomial having nth degree can have at most ‘n’ number of zeroes.

Ex : p(x) = xn -1 can have at most ‘n’ number of zeroes.

The process of a Quadratic polynomial p(x) = ax2 + bx + c (a ≠ 0) are the x-co-ordinates of the points where the graph intersect the x-axis

9) There are three types of graphs

Case I : p(x) = ax2 + bx + c

If discriminant = b2 - 4ac > 0 then ‘x’ has two real and distinct roots it seems graph intersects x axis in two points

Fig (a) Fig (b)

p(x) = x2 - 7x + 12 (a > 0)

= (x - 3)(x - 4)

graph intersects x-axis

(3,0) and (4,0)

p(x) = -x2 - x + 12 (a < 0)

= -x2 -4x + 3x +12

= -x(x+4) +3(x+4)

= (x+4)(3-x)

graph intersects x-axis at

(-4, 0) and (3,0)

Case II : p(x) = ax2 + bx + c

If discriminant b2 - 4ac < 0, then x has two imaginary distinct values it seems graph does n’t Intersect x-axis

Fig (C) Fig (D)

ex : x2 + x -7

ex : -x2 +2x -8

Case III : p(x) = ax2 + bx + c

If discriminant , then x has two real and equal roots (roots will co-incide) it seems graph touches x-axis at only one point

ex : x2 -6x +9

ex : -x2 +6x -9

10. Geometric meaning of the zeroes of polynomial

Case : I :

Let us take a linear polynomial y = p(x) = 3x +7

x 0 -1

y=3x+7 7 4

points (0,7) (-1,4)

∴ Graph of a polynomial is straight line passing through the point (-1, 4), (0, 7)

Case II: Let us consider a Quadratic polynomial

p(x) = x2 - 3x +4

x 1 2 3 4

y=x2 - 4x+3 0 -1 0 3

points (1,0) (2,-1) (3, 0) (4, 3)

∴ Graph of a polynomial is a parabola

Case III : Let us consider a cubic polynomial

p(x) = x3 - x2

Let p(x) = 0 => x3 - x2 = 0

=> x2(x-1) = 0

=> x = 0, x = 1

∴ The curve p(x) = x3 - x2 intersects x-axis at (0, 0) and (1, 0)

11) Types of polynomials :

i) Constant polynomial : A polynomial having degree ‘0’ is called constant polynomial

For ex: √11, 3, -7/8, 1 and so on

Degree of constant polynomial ‘0’ because

-7/8 ÷ -7/8 × 1 = -x0

ii) Zero polynomial : ‘0’ is called Zero polynomial. We can’t define degree of zero polynomial

0 = 0.xn + 0.xn-1 + .....

as ‘n’ may be any number. We can’t say degree

Degree is not defined

iii) Linear polynomial : A polynomial with degree one is called Linear polynomial, Graph of a linear polynomial is a straight line

For ex : 11x, 3x/2, √2/(x-1), 3x+1, 5p - 3, 6q -13/2 and so on

iv) Quadratic polynomial : A polynomial of degree ‘2’ is called Quadratic polynomial

For ex: 3x2 + 5x +7, x2 -5x +6, ax2 + bx + c (a ≠ 0)

12. Value of a polynomial : If p(x) is a polynomial at x = K. The Value of polynomial is defined as p(K), it may be zero or any non-zero real number

For ex : 1) For a polynomial p(x) = x2 -5x +6 at x = 2, p(2) = 2 -5.2 +6 = 0

∴ p(2) = 0

2) For a polynomial p(x) = x3 -7x2 +8x -6 -5 at x = 1

p(1) = 8.1 -7.1 +6.1 -5

= 8 -7 +6 -5 = 2

p(1) ≠ 0

13. Remainder Theorem

Let p(x) be a polynomial of degree greater than or equal to one, if p(x) is divided by (x - a). Then remainder R = p(a)

∴ p(a) may be zero (or) non-zero real number

Proof : If p(x) is divided by (x-a). Let we have obtained quotient Q(x) and remainder r(x)

∴ by divisior Algorithm

p(x) = (x-a)q(x) + r(x)

if x = a p(a) => r = p(a)

For ex : Let p(x) = x2 -3x +5, if p(x) is divided by x - 2

x-2 = 0

2 ∴ r = p(2) = 2 -3.2 +5 = 3 x = 2

∴ r = 3

Remarks :

Divisor Remainder

x-a f(a)

x+a f(a) -

ax+b f(b/a)

ax-b f(b/a)

Some Algebraic Identities

1. (a + b)2 = a2 +2ab +b2

2. (a - b)2 = a2 -2ab +b2

3. (a + b)2 - (a - b)2 = 4ab

4. (a + b)(a - b) = a2 -b2

5. (a + b)(b - a) = -a2 +b2

6. (a + b)3 = a3 +3a2b +3ab2 +b3

7. (a - b)3 = a3 -3a2b +3ab2 -b3

8. a3 +b3 = (a+b)(a2 -ab +b2)

9. a3 -b3 = (a-b)(a2 +ab +b2)

10. (a + b + c)2 = a2 +b2 +c2 +2ab +2bc +2ca

11. a3 +b3 +c3 -3abc = (a+b+c)(a2 +b2 +c2 -ab -bc -ca)

12. If a3 +b3 +c3 = 3abc => a+b+c = 0

13. (a - b)(a - b) ... = ...

Special Products

1. (x+a)(x+b) = x2 +(a+b)x +ab

2. (ax+b)(cx+d) = acx2 +(ad+bc)x +bd

3. (x+a)(x+b)(x+c) = x3 +(a+b+c)x2 +(ab+bc+ca)x +abc

Factor Theorem : Let p(x) be a polynomial of any degree if R = p(K) = 0 , then (x-k) is said to be factor of p(x)

ex : Let p(x) = x2 -5x +6 According factor theorem x = 2 (2)

consider p(2) = 2 -5.2 +6 is a factor of p(x)

= 4 -10 +6

p(2) = 0

Reamarks:

1) If (x - 1) is a factor of f(x) = a0 + a1x + a2x2 + ..... + anxn then sum of the coefficient is equal to zero.

i.e., a1 + a2 + a3 + ..... + an = 0

Explanetion : ∵(x - 1) is a factor of f(x)

f(1) = 0

a0 + a1.1 + a2.1 + .......... = 0

0 + a1 + a2 + .......... = 0

∴ sum of the coefficents = 0

2) If (x + 1) is a factor of f(x) , then sum of the coefficients of even powers of x = sum of the coefficients of odd powers of x

i.e., a0 + a2 + a4 + .......... = a1 + a3 + a5 + ........

Explanetion : ∵(x + 1) is a factor of f(x)

f(-1) = 0

i.e., a0 - a1 + a2 - a3 + .......... = 0

a0 + a2 + a4 + .......... = a1 + a3 + a5 + ........

Note: Constant is taken as coefficient of even power of x.

3) xn - yn is divisible by x - y for every positive integers ‘n’

For ex : 1) x2 - y2 is divisible by (x - y) 2) x3 - y3 is divisible by (x - y)

4) xn - yn is divisible by (x+y) for every positive even Integer n

For ex: x4 - y4 is divisible by (x+y)

5) xn + yn is divisible by (x+y) for every odd positive Integer ‘n’

For ex : x3 + y3 is divisible by (x+y)

Synthetic division of Horner’s method

Horner’S Method of synthetic deivision :

We sheall explain the method with the following examples :

To divide x4 + 4x3 + 3x2 - 4x - 4 by (x-1)

(Multiplier = )

1 4 3 4 4

0 1 5 8 4

1

_______________

1 5 8 4 0

- -

The quotient is x3 + 5x2 + 8x +4

Explanation :

First horizontal row contains the multiplier which is obtained by the zero of x - 1, which is 1.

The remaining elements in the first horizontal row are the coefficients of descending power of x (The coefficinet of missing power of x, if any, that should be taken as zero).

To form the second horizontal row start with zero right under the second element of first row and add, the result is 1 which is first entry in the third row.

Multiply this 1 with the multiplier 1 put the result under 4 (the third entry of the first row). Thus we get the second entry 1 in the second row. Then add 4 and 1 to get the second entry in the third row which is 5. Now multiply this 5 with the multiplier, put the product right under 3(the fourth entry of the first row). Add 3 amd 5 to get third entry in the third row. Thus the third entry in the third row is 8. Now multiply 8 with the multiplier 1, put the product under -4 (the fifth entry of th first row). Add 8 and -4 to get the fourth entry in the third row. Repeat the same procedure to get zero as the fifth entry of the third row. The last entry in the third row stands for the remainder, while the first four figures stand for the coefficients of descending powers of x of quotient.

Thus the quotient is x3 + 5x2 + 8x + 4. We shall return to our problem.

f(x) = x4 + 4x3 + 3x2 - 4x - 4 = (x-1)(x3 +5x2 +_8x +4)

Now if we write g(x) = x3 -5x2 +8x +4

g(-1) = (-1)3 + 5(-1)2 + 8(-1) +4 = 0

∴ (x+1) is a factor of g(x). Hence a factor of f(x).

Now to divide g(x) by (x+1), the multiplier is -1, the zero of x + 1.

Let us once again apply synthetic division, to g(x).

1 5 8 4

0 1 4 4

1

__________________

1 4 4 0

- - -

-

The quotient is x2 + 4x + 4 = 0

∴ f(x) = (x-1) g(x) = (x-1)(x+1)(x2 +4x +4) = (x-1)(x+1)(x+2)2

H.C.F and L.C.M of Polynomials :

If a polynomial p(x) is a product of two polynomials h(x) and g(x) i.e., f(x) = g(x) × h(x) then g(x) and h(x) are said to be factor of x.

Ex : Let f(x) = x2 -7x +10

f(x) = (x -2)(x -5)

Note : If h(x) is a factor of f(x)

∴ -h(x) is a factor of f(x).

Highest common factor (H,C,F) or Greatest common divisior (G.C.D):

The product of the least powers of the common factors is said to be H.C.F of the given polynomials.

Let f(x) = x3 - x2 - x -1 = (x -1)(x -2)(x -3)(x -4)

g(x) = x3 - x = (x -1)(x -2)(x -3)

∴ H.C.F = (x -1) × 1 × 1 × 1 = (x -1)

Least common multiple of polynomials (L.C.M)

The product of the highest powers of common factors is said to be L.C.M of the given polynomials.

Let f(x) = (x -1)(x -2)(x -4)

g(x) = (x -1)(x -2)(x -4)

∴ L.C.M = (x -1)(x -2)(x -4)

WORK SHEET - I

1. Which of the following is not a polynomial

A) x3 - x + 2

B) x + 1/x + 3

C) x2 + 3/x

D) x2 - 1/5x

2. Which of the following is correct

A) All Algebraic expressions are Polynomials

B) All Polynomials are Algebraic expressions

C) All Polynomials are not Algebraic expressions

D) None of these

3. The degree of (2x5 - 11x3 + x4)/3

A) 7 B) 10 C) 9 D) -11

4. The degree of 5x3 + 4x2y - 13z

A) 8 B) 11 C) -10 D) 9

5. If x2 - 2x + 1 = Ax2 + Bx + c, then what is A/B

A) 1 B) -1 C) D) 0

6. If p(x) = x2 -3x +2, clearly a + b + c = 0 , then

A) a3 +b3 +c3 = 3abc

B) a3 +b3 +c3 = 3(a+b+c)

C) a3 +b3 +c3 = 3(a+b+c)

D) All

7. The degree of the product cubic polynomial and a biquadratic polynomial

A) 9 B) 7 C) 14 D) 12

8. Degree of constant polynomial

A) constant B) 1 C) 0 D) any real number

9. Zero of the polynomial 2x+1

A) -2 B) -1 C) -1/2 D) none of these

10. If 2ax + bx + c is a quadratic polynomial

A) a = 0 B) a ≠ 0 C) a > 0 D) a < 0

JEE Mains

MCQ’s with single correct answers type

1. Factors of x2 + xy + yz - z2 are

A) (x+y+z) (x-y+z) B) (x+y+z) (x-z+y) C) (x+y+z) (x-y+z) D) (y+z-x) (y+z+x)

2. If P(x) = 3x2-5x, then the zero of this polynomial

A) 3/0, 5 B) 5/0, 3 C) 3/0, 5 C) 5/0, 3

3. Then number of zero of the polynomial having degree ‘n’

A) Atmost ‘n’ number of zeroes B) Atleast ‘n’ number of zeroes.

C) (n+1) number of zeroes. D) (n-1) number of zeros

4. The remainder when x3 + px2 + 6 is divided by x - p

A) 5+p B) p3 C) 5 p D) 5 - p

5. The remainder when f(x) = ax3 + bx2 + cx + d is divided (x - 1)

A) a+b+c-d B) a+b = c+d C) a+b+c+d D) a+c = b+d

6. If both (x - 2) and (1/2 x -) are the factors of 2px + x + 5, then

A) p2 - r2 = 0 B) p2 + r2 = 0 C) p2.r3 = 0 D) p2 + r2 = 0

7. If 1/3 a + 1/3 b + 1/3 c = 0 , then (a+b+C)3

A) 54abc B) 81abc C) 27abc D) 72abc

8. If a+b+c = 9 and ab+bc+ca = 2b, then a2 + b2 + c2.

A) 28 B) 27 C)29 D)31

9. Factors of (1/4 x - 5/4 y)(1/4 x - 5/4 y) are

A) 2(2 5 ) x y + B) 2(5 2 ) y x - C) 2(5 2 ) y x + D) 2(5 2 ) x y -

10. Factors of 12 4 4 12 a x a x - are

A) 4 4 4 4 2 2 a x a x a x a x a x ( )( )( )( ) + + + -

B) 4 4 4 4 2 2 a x x a a x a x a x ( )( )( )( ) - + + -

C) 4 4 4 4 2 2 a x a x x a a x a x ( )( )( )( ) + - + -

D) 4 4 6 6 a x a x a x a x ( )( )( ) - + -

11. Factors of 1-2cb - (a2-b2) are

A) (1+a+b) (1-a+b) B) (1-a+b) (1-a-b) C) (1+ab) (1+a+b) D) (1-a-b) (a+b)

12. If the length and breadth are the factors of a rectangle whose area is given by p(y) 35y2+13y -12, then l and b are

A) 7y - 3, 5y +4 B) 7y + 3, 5y - 4 C) 7y + 4, 5y + 3 D) 7y - 4, 5y+3

13. The number of rational factors of x12 - y12

A)7 B)8 C)6 D)10

14. One of the factors of a3 -b3+1+3ab

A)(1) a b + + B)(1) a b - + C)(1) a b + - D) (1) a b - -

15. One of the factors of p3(q-r)3+q3(r-p)3+r3(p-q)3

A)-pqr B) (p+q+r) C) (p)(q)(q)(r)(r)(p) + + - D) pqr

16. If 3x = a+b+c, then the value of (x - a)3 +(x - b)3+ (x - c)3.

A) 3(x-a)(x-b)(x-c) B) 3(x+a)(x+b)(x+c)

C) 3(a-x)(x-b)(x-c) D) 3(x-a)(x-b)(c-x)

17. The factors of

(a-b)3 +(b-c)3 +(c-a)3

A) (a+b) (b-c) (c+a) B) (a+b) (b+c) (c+a)

C) (b-a) (c-b) (a-c) D) (abc+ab+bc+ca

18. Which of the following expression value can be foud by using, if a+b+c = 0, then

a3+b3+c3 = 3abc

A) 303+203+503 B)-303+203+503

C) 303+203-503 D) 303+203+503

19. If f(x)= x4-2x3+3x2-ax-b, when divided by x - 1, the remainder is 6 then a+b

(A) 4 B) 5 C) -4 D) 0

20. If x140+2.x151+k is divisible by x+1, then the value of k is

A)1 B)-3 C)2 D) -2

21. If (3x-1)7= a7.x7+a6.x6+a5.x5+..........+a1x+a0, then a0+a1+a2+a3+....+a7 =

A) 0 B) 1 C) 128 D) 64

22. The polynomials ax3+3x2-13 and 2x3-5x+a are divided by x+2 if the remainder in each case is the same what is a

(A) 9/5 B) -9/5 C) -5/9 D) 5/9

23. If a/b = b/c then product of the factors (a+b+C) (a-b+C)

A) a2+c2-ac B) a2 +c2+ac C) a2 - c2-ac D) c2-a2+ac

24. The L.C.M of xy+yz+zx+y2 and x2+xy+yz+zx is

A) (x+y) (y+z) B) (x+y) (y+z) (z+x) C) (y+z) (z+x) D) (x+y) (z+x)

25. The factors of (x+y)(1-z) - (y+z) (1-x) =

A)(x-z) (1-y) B)(x-z) (1-z) C)(x+y) (1-y) D)(x-z) (1+y)

26. If x4+x3 is divide by x+9, then the degree of the remainder

A) 1 B) 0 C) 2 D) 3

27. The remainder when x3+3x2+3x+1 is exactly divisible by x - π, if π = 22/7

A) 3/27/7 B) 3/30/7 C) 3/29/7 D) 3/32/7

28. What must be added to 3x3+x2-2x+9 so that the result is exactly divisible by 3x2+7x-6.

A) 2x-3 B) 2+3x C) 3x-2 D) 2x+3

29. If x2-1 is a factor of ax4+bx3+cx2+dx+e then

A) a+c+e = 0 B) b+d = 0 C) a+b+c+d+e = 0 D) All

30. If f(x) = cx2+d2 then, then zero of f(x)

A) -d/c B) -d/c C) -d2/c2 D) c2/d2

JEE Advanced

Multi correct answers type

1. If (x-a) (x-b) is a factor of a polynomial p(x)

A) p(a) = 0 B) p(ab) = 0 C) p(b) = 0 D) both b & c

2. If p = r which are the factors of p(x) = px2+5x+r

A) x+2 B) x - 1/2 C) x +1/2 D) x -2

3. If (x2-1) is a factor of p(x)=a0.xn + a1.xn-1 + a2.xn-2 + ...... + an then

A) a0+a1+......a = 0 B) a0+a2+.......an =0

C) a0+a3+a5...+an-1 = 0 D) a0+a2+a4+......= a1+a3+a5+........

4. If x=2 and x = 0 are the roots of the polynomial f(x) = 2x3-5x2+ax+b then the values of a & b

A) a = 2 B) b = 0 C) b = -2 D) ab = -4

5. If (x2-1) is a factor of ax4+bx3+cx2+dx+e then

A)a+c+e = b+d B) a+b+e+ = c+d C) a+b+c+d+e=0 D) b+c+d = a+e

Reasoning type

A) both statement I & II are true.

B) both statement I & II are false

C) Statement I is true but statement II is false

D) Statement I is false but statement II is true

6. Statement I: (x - 2) is the factor of the expression x3+ax2+bx+6 when this expression is divided by x-3, it leaves remainder 3 and a2+b2 is 10.

Statement II : Let p(x) be a polynomial of degree greater than or equal to 1 and a be a real number such that p(A) = 0, then (x -A) is a factor of p(x)

7. Statement -I: The factors of p(x) x3-6x2+11x-6 are (x-1), (x-2) & (x-3)

Statement-II: If α, β, γ are the zeroes of p(x) = ax3+bx2+cx+d then (x-α), (x-β) and (x-γ) are the zeroes.

8. Statement I : If (3x-1)7 = a7x7+a6.x6+a5.x5+.....+a1x+a0 then sum of coefficient is 128.

Statement II: By substituting x =-1, we get the sum of the coefficient.

Comprehension

Paragraph I : If (ax+B)is a factor of p(x), then its remainder

9. If 3x-1 is a factor of p(x)= x2+ax, then value of ‘a’

A) 1/3 B) 3 C) -1/3 D) -1/2

10. If x-1 is a factor of p(x)= x3-3x2+3x-1 then p(1)+p(-1) =

A) 0 B) 8 C)-8 D) none of these

11. If (x-1) and (x-2) are the factors of p(x)= ax+b then a+b =

A)3 B) -1 C)0 D) 1

Paragraph II :

If p(A) = 0, p(B) = 0, p(C) = 0, then the polynomial is p(x)=(x-A) (x-B) (x-C)

12. If p(1/2) = 0, p(√2) = 0 then the polynomial p(x)

A) x2 + (√2 +1/2)x + 1/√2

B) x2 + (2 +1/√2)x + 1/√2

C) x2 - (√2 +1/2)x + 1/√2

D) x2 - 5/2 x + 1/2

13. If p(m/n) = 0 and p(-m/n) = 0, then p(x) =

A) n2x2+m2 B) m2x2-n2 C) n2x2-m2 D) m2n2+x2

14. If p(√2) = 0 and p(√3) = 0 then product of the zeroes of the polynomial

A) √6 B) -√6 C) √(2/3) D) √(3/2)

Paragraph III : If a+b+c = 0 then a3+b3+c3 =3abc

15. If x-y+z = 0, then x3+z3

A) 3xyz-y3 B) y3+3xyz C) 3y3+xyz D) y3-3xyz

16. If 1/3 x + 1/3 y + 1/3 z = 0 , then (x+y+z)3

A)9xyz B)27xyz C)81xyz D) 3xyz

17. If l-m-n = 0 then l3=

A)3lmn-m3-n3 B)3lmn+m3n3 C)3lmn - m3+n3 D)m3+n3-3lmn

Integer Answer Type :

18. If +b+c = 8, ab+bc+ca = 26 then the value of a2+b2+c2-20 = ______

19. If (x-1) and (x+2) are both the factors of expression x3-ax2+bx-10, then a/b = ______

20. The zero of the polynomial x3-23x2+142x-120 =__________

21. _________

Matrix Match Type

22. Match the following with list I to list II

List - I

A) x2y2 + 2xyz2 - x2z2 - y2z2

B) x3 + y3 - z3 + 3xyz

C) (y-z)x3 + (z-x)y3 + (x-y)z3

D) x2(y-z) + y2(z-x) + z2(x-y) + xyz

List - II

p) x3 + xyz + (y3 - z3)

q) x2(y-z) + (y2 + z2 + yz)x - y2z - z2y

r) (y2 - z2)x2 + 2yz2x - y2z2

s) (y-z)x3 - y3x + z3x + yz3 - yz3

t) (y-z)x3 - y3x + z3x - yz2 + yz3

23. List - I

a) a3(b-c)+b3(c-a)+c3(a-b)

b) a4(b-c)+b4(c-a)+c4(a-b)

c) x3(y-z)+y3(z-x)+z3(z-y)

d) x4(y-z)+y4(z-x)+z4(z-y)

List - II

p) (x-y) (y-z) (z+x)

q) (a-b) (b-c) (c-a)

r) (x-y) (y-z) (z-x)

s) (a-b) (b-c) (c+a)

t ) (x+y) (y-z) (z-x)