#### Chapters

Chapter 2: Banking (Recurring Deposit Account)

Chapter 3: Shares and Dividend

Chapter 4: Linear Inequations (In one variable)

Chapter 5: Quadratic Equations

Chapter 6: Solving (simple) Problems (Based on Quadratic Equations)

Chapter 7: Ratio and Proportion (Including Properties and Uses)

Chapter 8: Remainder and Factor Theorems

Chapter 9: Matrices

Chapter 10: Arithmetic Progression

Chapter 11: Geometric Progression

Chapter 12: Reflection

Chapter 13: Section and Mid-Point Formula

Chapter 14: Equation of a Line

Chapter 15: Similarity (With Applications to Maps and Models)

Chapter 16: Loci (Locus and Its Constructions)

Chapter 17: Circles

Chapter 18: Tangents and Intersecting Chords

Chapter 19: Constructions (Circles)

Chapter 20: Cylinder, Cone and Sphere

Chapter 21: Trigonometrical Identities

Chapter 22: Height and Distances

Chapter 23: Graphical Representation

Chapter 24: Measure of Central Tendency(Mean, Median, Quartiles and Mode)

Chapter 25: Probability

## Chapter 8: Remainder and Factor Theorems

#### Exercise 8(A) [Pages 108 - 109]

### Selina solutions for Concise Mathematics Class 10 ICSE Chapter 8 Remainder and Factor Theorems Exercise 8(A) [Pages 108 - 109]

Find , in given case, the remainder when :

`x^4-3x^2+2x+1` is dividend by x-1

Find, in given case, the remainder when:

`x^3+3x^2-12x+4` is divided by x-2

Find , in given case the remainder when:

`x^4+1` is divided by x+1

show that

`x-2` is a factor of `5x^2+15x-50`

show that

`3x+2` is a factor of `3x^2-x-2`

Use the Remainder Theorem to find which of the following is a factor of 2x^{3} + 3x^{2} – 5x – 6.

x + 1

Use the Remainder Theorem to find which of the following is a factor of 2x^{3} + 3x^{2} – 5x – 6.

2x – 1

Use the Remainder Theorem to find which of the following is a factor of 2x^{3} + 3x^{2} – 5x – 6.

x + 2

If 2x + 1 is a factor of 2x^{2} + ax – 3, find the value of a.

Find the value of k, if 3x – 4 is a factor of expression `3x^2` + 2x − k.

Find the values of constants a and b when x – 2 and x + 3 both are the factors of expression x^{3} + ax^{2} + bx – 12.

Find the value of k, if 2x + 1 is a factor of (3k + 2)x^{3} + (k − 1)

Find the value of a, if x – 2 is a factor of 2x^{5} – 6x^{4} – 2ax^{3} + 6ax^{2} + 4ax + 8.

Find the values of m and n so that x – 1 and x + 2 both are factors of x^{3} + (3m + 1) x^{2} + nx – 18.

When `x^3 + 2x ^2– kx + 4 `is divided by x – 2, the remainder is k. Find the value of constant k.

Find the value of a, if the division of ax^{3} + 9x^{2} + 4x – 10 by x + 3 leaves a remainder 5.

If x^{3} + ax^{2} + bx + 6 has x – 2 as a factor and leaves a remainder 3 when divided by x – 3, find the values of a and b.

The expression 2x^{3} + ax^{2} + bx – 2 leaves remainder 7 and 0 when divided by 2x – 3 and x + 2 respectively. Calculate the values of a and b

What number should be added to 3x^{3} – 5x^{2} + 6x so that when resulting polynomial is divided by x – 3, the remainder is 8?

What number should be subtracted from x^{3} + 3x^{2} – 8x + 14 so that on dividing it with x – 2, the remainder is 10.

The polynomials 2x^{3} – 7x^{2} + ax – 6 and x^{3} – 8x^{2} + (2a + 1)x – 16 leaves the same remainder when divided by x – 2. Find the value of ‘a’.

If (x – 2) is a factor of the expression 2x^{3} + ax^{2} + bx – 14 and when the expression is divided by (x – 3), it leaves a remainder 52, find the values of a and b

Find ‘a‘ if the two polynomials ax^{3} + 3x^{2} – 9 and 2x^{3} + 4x + a, leave the same remainder when divided by x + 3.

#### Exercise 8(B) [Pages 111 - 112]

### Selina solutions for Concise Mathematics Class 10 ICSE Chapter 8 Remainder and Factor Theorems Exercise 8(B) [Pages 111 - 112]

Using the Factor Theorem, show that:

(x – 2) is a factor of x^{3} – 2x^{2} – 9x + 18. Hence, factorise the expression x^{3} – 2x^{2} – 9x + 18 completely.

(x + 5) is a factor of 2x^{3} + 5x^{2} – 28x – 15. Hence, factorise the expression 2x^{3} + 5x^{2} – 28x – 15 completely.

(3x + 2) is a factor of 3x^{3} + 2x^{2} – 3x – 2. Hence, factorise the expression 3x^{3} + 2x^{2} – 3x – 2 completely.

Using the Remainder Theorem, factorise each of the following completely.

3x^{3 }+ 2x^{2} − 19x + 6

Using the Reminder Theorem, factorise of the following completely.

2x^{3} + x^{2} – 13x + 6

Using the Remainder Theorem, factorise each of the following completely.

3x^{3} + 2x^{2} – 23x – 30

Using the Remainder Theorem, factorise each of the following completely.

4x^{3} + 7x^{2} – 36x – 63

Using the Remainder Theorem, factorise each of the following completely

x^{3} + x^{2} – 4x – 4

Using the Remainder Theorem, factorise the expression 3x^{3} + 10x^{2} + x – 6. Hence, solve the equation 3x^{3} + 10x^{2} + x – 6 = 0.

Factorise the expression f (x) = 2x^{3} – 7x^{2} – 3x + 18. Hence, find all possible values of x for which f(x) = 0.

Given that x – 2 and x + 1 are factors of f(x) = x^{3} + 3x^{2} + ax + b; calculate the values of a and b. Hence, find all the factors of f(x).

The expression 4x^{3} – bx^{2} + x – c leaves remainders 0 and 30 when divided by x + 1 and 2x – 3 respectively. Calculate the values of b and c. Hence, factorise the expression completely.

If x + a is a common factor of expressions f(x) = x^{2} + px + q and g(x) = x^{2} + mx + n;

show that : `a=(n-q)/(m-p)`

The polynomials ax^{3} + 3x^{2} – 3 and 2x^{3} – 5x + a, when divided by x – 4, leave the same remainder in each case. Find the value of a.

Find the value of ‘a’, if (x – a) is a factor of x^{3} – ax^{2} + x + 2.

Find the number that must be subtracted from the polynomial 3y^{3} + y^{2} – 22y + 15, so that the resulting polynomial is completely divisible by y + 3.

#### Exercise 8(C) [Page 112]

### Selina solutions for Concise Mathematics Class 10 ICSE Chapter 8 Remainder and Factor Theorems Exercise 8(C) [Page 112]

Show that (x – 1) is a factor of x^{3} – 7x^{2} + 14x – 8. Hence, completely factorise the given expression.

Using Remainder Theorem, factorise:

x^{3} + 10x^{2} – 37x + 26 completely

When x^{3} + 3x^{2} – mx + 4 is divided by x – 2, the remainder is m + 3. Find the value of m.

What should be subtracted from 3x^{3} – 8x^{2} + 4x – 3, so that the resulting expression has x + 2 as a factor?

If (x + 1) and (x – 2) are factors of x^{3} + (a + 1)x^{2} – (b – 2)x – 6, find the values of a and b. And then, factorise the given expression completely.

if x – 2 is a factor of x^{2} + ax + b and a + b = 1, find the values of a and b.

Factorise x^{3} + 6x^{2} + 11x + 6 completely using factor theorem.

Find the value of ‘m’, if mx^{3} + 2x^{2} – 3 and x^{2} – mx + 4 leave the same remainder when each is divided by x – 2

The polynomial px^{3} + 4x^{2} – 3x + q is completely divisible by x^{2} – 1; find the values of p and q. Also, for these values of p and q factorize the given polynomial completely.

Find the number which should be added to x^{2} + x + 3 so that the resulting polynomial is completely divisible by (x + 3).

When the polynomial x^{3} + 2x^{2} – 5ax – 7 is divided by (x – 1), the remainder is A and when the polynomial x^{3} + ax^{2} – 12x + 16 is divided by (x + 2), the remainder is B. Find the value of ‘a’ if 2A + B = 0.

(3x + 5) is a factor of the polynomial (a – 1)x^{3} + (a + 1)x^{2} – (2a + 1)x – 15. Find the value of ‘a’, factorise the given polynomial completely.

When divided by x – 3 the polynomials x^{3} – px^{2} + x + 6 and 2x^{3} – x^{2} – (p + 3) x – 6 leave the same remainder. Find the value of ‘p’.

Using the Remainder Theorem, factorise each of the following completely.

2x^{3} + x^{2} – 13x + 6

Using remainder theorem, find the value of k if on dividing 2x^{3} + 3x^{2} - kx + 5 by x - 2, leaves a remainder 7

What must be subtracted from 16x^{3} – 8x^{2} + 4x + 7 so that the resulting expression has

2x + 1 as a factor?

## Chapter 8: Remainder and Factor Theorems

## Selina solutions for Concise Mathematics Class 10 ICSE chapter 8 - Remainder and Factor Theorems

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