# Free Polynomials 01 Practice Test - 10th Grade

### Question 1

Which of the following is a polynomial in 1 variable?

4x + 2

1(x+1)

4x + 2 = 0

x + y

#### SOLUTION

Solution :A

A polynomial is a mathematical expression of one or more algebraic terms each of which consists of a constant multiplied with one or more variables raised to a non-negative integral power . For example, (a+bx+cx2) where a, b and c are constants, and x is the variable.

Degree is the highest power of the variable.

4x+2 is a polynomial of degree 1.

1(x+1)=(x+1)−1, the degree of this expression would not be non-negative. Thus, it is not a polynomial.

4x+2=0 is an equation and hence not a polynomial.

x+y is a polynomial in two variables, x and y.

### Question 2

Which of the following is not a polynomial?

x+x2

x+√3

3√y+y2

x

#### SOLUTION

Solution :C

All the variables of a polynomial should have whole numbers as exponents.

For x+x2,the exponents of the variable x are 1 and 2. All the exponents are whole numbers. Therefore, it is a polynomial.

In the expression 3√y+y2,exponent of 3√y is 13, but 13 is not a whole number. Hence, 3√y+y2 is not a polynomial.

For x+√3,the exponent of the variable is 1 which is a whole number. Therefore, it is a polynomial.

For x, the exponent of the variable is 1 which is a whole number. Therefore, it is a polynomial.

### Question 3

If p(x)=x2−3√3x+1, then the value of p(3√3) is 0.

True

False

#### SOLUTION

Solution :B

Given,p(x)=x2−3√3x+1x=3√3

p(3√3)=(3√3)2−3√3(3√3)+1p(3√3)=27−27+1p(3√3)=0+1=1

Thus, the given statement is false.

### Question 4

The zero of p(x) = 5x - 75 is

#### SOLUTION

Solution :To find the zeros of a polynomial expression, we must equate it to zero.

∴5x−75=0

⇒x=755

⇒x=15

### Question 5

What is the remainder when 3x2−7x+5 is divided by (x−1) ?

0

1

2

3

#### SOLUTION

Solution :B

Given f(x) = 3x2−7x+5

To find the remainder when it is divided by (x−1), we use remainder theorem.Remainder of f(x)(x−a) is f(a).

f(1)=3−7+5=1

⇒ The remainder when 3x2−7x+5 is divided by (x−1) is 1.

### Question 6

What is the coefficient of z in (z−5)3?

125

-25

5

75

#### SOLUTION

Solution :D

(z−5)3=z3−53−3×z2×5+3×z×52

[Using (a−b)3=a3−b3−3a2b+3ab2]

⇒(z−5)3 = z3−125−15z2+75z

Hence, the coefficient of z is 75.

### Question 7

Which of the following is a factor of

(x+y)3−(x3+y3)?

x2−2xy+y2

xy2

3xy

x2+xy+y2

#### SOLUTION

Solution :C

Let P(x)=(x+y)(x+y)2−(x+y)(x2−xy+y2)

=(x+y)(x2+2xy+y2)−(x+y)(x2−xy+y2)

=(x+y)(x2+2xy+y2−x2+xy−y2)

[Taking (x+y) as common]

=(x+y)(3xy)

Hence, from the given options 3xy is a factor of the given polynomial.

### Question 8

Which of the following are polynomials?

y3+4y

√2x−1

x+1x−1

z3+z

#### SOLUTION

Solution :B and D

The exponents of the variables of a polynomial should be a whole number.

In y3+4y=y3+4y−1, the exponent of y in (4y−1) is -1 which is not a whole number.

In √2x−1, the exponent of x is a whole number.

In x+1x−1=(x+1)×(x−1)−1, the exponent of x in (x+1)−1 is not a whole number.

In z3+z, the exponents of z in both the terms is a whole number exponent.

Hence, z3+z and √2x−1 are polynomials .

### Question 9

Which of the following is a zero of (49x2−1)+(1+7x)2 ?

#### SOLUTION

Solution :A

Given,(49x2−1)+(1+7x)2=((7x)2−12)+(1+7x)2=(7x−1)(7x+1)+(1+7x)(1+7x)=(1+7x)(7x−1+1+7x)[Taking (1+7x) common]=(1+7x)(14x)

To find the zeroes of a polynomial expression, we must equate it to zero.(1+7x)(14x)=0⇒(1+7x)=0; (14x)=0∴x=−17,0Zeroes are −17 & 0.

### Question 10

A polynomial y=f(x) when represented on graph cuts x−axis at 2 points and y−axis at 3 points. What is the number of zeroes of f(x) ?

#### SOLUTION

Solution :D

The zeroes of a polynomial are the points which cuts the x−axis on the graph. As f(x) cuts x−axis at 2 points, the number of zeroes are 2.