Frank solutions for Mathematics - Part 2 [English] Class 10 ICSE chapter 10 - Remainder And Factor Theorems [Latest edition]

Find without division, the remainder in the following: 

5x² - 9x + 4 is divided by (x - 2) 

Find without division, the remainder in the following: 

5x³ - 7x² +3 is divided by (x-1)  

Find without division, the remainder in the following: 

8x² - 2x + 1 is divided by (2x+ 1) 

Find without division, the remainder in the following :

x³ + 8x² + 7x- 11 is divisible by (x+4)

Find without division, the remainder in the following: 

2x³ - 3x² + 6x - 4 is divisible by (2x-3) 

Prove by factor theorem that 

(x-2) is a factor of 2x³- 7x -2 

Prove by factor theorem that 

(2x+1) is a factor of 4x³ + 12x² + 7x +1 

Prove by factor theorem that 

(3x-2) is a factor of 18x³ - 3x² + 6x -12  

Prove by factor theorem that 

(2x - 1) is a factor of 6x³ - x² - 5x +2 

Prove by factor theorem that 

(x - 3) is a factor of 5x² - 21 x +18 

Find the values of a and b in the polynomial f(x) = 2x³ + ax² + bx + 10, if it is exactly divisible by (x+2) and (2x-1).

Using remainder theorem, find the value of m if the polynomial f(x)= x³ + 5x² -mx +6 leaves a remainder 2m when divided by (x-1),  

Find the value of m ·when x³ + 3x² -m x +4 is exactly divisible by (x-2) 

Find the values of p and q in the polynomial f(x)= x³ - px² + 14x -q, if it is exactly divisible by (x-1) and (x-2). 

Find the values of a and b when the polynomial f(x)= ax³ + 3x² +bx -3 is exactly divisible by (2x+3) and leaves a remainder -3 when divided by (x+2). 

Find the values of m and n when the polynomial f(x)= x³ - 2x² + m x +n has a factor (x+2) and leaves a remainder 9 when divided by (x+1).

Find the values of a and b when the polynomials f(x)= 2x² -5x +a and g(x)= 2x² + 5x +b both have a factor (2x+1).

Find the values of a and b when the factors of the polynomial f(x)= ax³ + bx² + x a are (x+3) and (2x-1). Factorize the polynomial completely.

What number should be subtracted from x² + x + 1 so that the resulting polynomial is exactly divisible by (x-2) ? 

What number should be added to 2x³ - 3x² + 7x -8 so that the resulting polynomial is exactly divisible by (x-1) ?

What number should be subtracted from the polynomial f(x)= 2x³ - 5x² +8x -17 so that the resulting polynomial is exactly divisible by (2x - 5)?

What number should be added to polynomial f(x)= 12x³ + 16x² - 5x - 8 so that the resulting polynomial is exactly divisible by (2x - 1) ?

use the rernainder theorem to find the factors of ( a-b )³ + (b-c )³ + ( c-a)³ 

Prove that ( p-q) is a factor of (q - r)³ + (r - p) ³ 

Prove that (x - y) is a factor of yz( y² - z²) + zx( z² - x²) + xy ( x² - y²) 

Prove that (x-3) is a factor of x³ - x² - 9x +9 and hence factorize it completely. 

Prove that  (x+ 1) is a factor of x³ - 6x² + 5x + 12 and hence factorize it completely.

Prove that (5x - 4) is a factor of the polynomial f(x) = 5x³ - 4x² - 5x +4. Hence factorize It completely.

A polynomial f(x) when divided by (x - 1) leaves a remainder 3 and when divided by (x - 2) leaves a remainder of 1. Show that when its divided by  (x - i)(x - 2), the remainder is (-2x + 5). 

The polynomial f(x) = ax⁴ + x³ + bx² - 4x + c has (x + 1), (x-2) and (2x - 1) as its factors. Find the values of a,b,c, and remaining factor.