Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 8

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Given that $f (x) = x^{2} + a x + b$ is a polynomial with integral coefficients. If $f$ is a common factor of polynomials $g (x) = x^{4} - 3 x^{3} + 2 x^{2} - 3 x + 1$ and $h (x) = 3 x^{4} - 9 x^{3} + 2 x^{2} + 3 x - 1$ , find $f (x)$ .

Important Questions on Algebra

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 9

For any non-negative integers

m, n, p,

prove that the polynomial

x^{3 m} + x^{3 n + 1} + x^{3 p + 2}

has the factor

x^{2} + x + 1 .

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 10

When

f (x) = x^{3} + 2 x^{2} + 3 x + 2

is divided by

g (x)

which is a polynomial with integer coefficients, the quotient and remainder are both

h (x)

. Given that

h

is not

a

constant, find

g

and

h

.

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 11

The graph of the function $f$ is shown below. How many solutions does the equation $f (f (x)) = 6$ have?

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 15

Find an integer

x

that satisfies the equation

x^{5} - 101 x^{3} - 999 x^{2} + 100900 = 0

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 18

Let

f (x) = a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{n} x^{n}

, where

a_{i}

are non-negative integers for

i = 0, 1, 2, \dots, n

. If

f (1) = 21

and

f (25) = 78357

, find the value of

(\frac{f (10) + 3}{100})

.

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 20

There are a few integer values of

a

such that

\frac{a^{2} - 3 a - 3}{a - 2}

is an integer. Find the sum of all these integer values of

a

.

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 22

Let

a

and

b

be positive real numbers such that

\frac{1}{a} - \frac{1}{b} - \frac{1}{a + b} = 0

. Find the value of

{(\frac{b}{a} + \frac{a}{b})}^{2}

.

EASY

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Non-Routine Mathematics Resource Book-1 for PRMO>Chapter 1 - Algebra>EXERCISE - 1.2>Q 23

Let the sum of the coefficients of the polynomial ${(4 x^{2} - 4 x + 3)}^{4} {(4 + 3 x - 3 x^{2})}^{2}$ is $S$ . Find $\frac{S}{16}$ .