Fundamental Concepts of Algebra — Multiple Choice Questions

Coefficient of $x$ in $−xyz$ is:

1. $−xyz$

2. $−yz$

3. $yz$

4. $−xy$

Answer

Any factor or group of factors of a product is the coefficient of the remaining factors.

$-xyz = (-yz) \times x$, so the coefficient of $x$ in $-xyz$ is $-yz$.

Hence, option 2 is the correct option.

Numeral coefficient in $−8x \div 7z$ is:

1. $−56$

2. $-\dfrac{7}{8}$

3. $-\dfrac{8}{7}$

4. $−15$

Answer

If a factor of a product is a numerical quantity, it is called the numeral coefficient of the remaining factors.

Solving,

$\Rightarrow -8x \div 7z \\[1em] \Rightarrow \dfrac{-8x}{7z} \\[1em] \Rightarrow \left(\dfrac{-8}{7}\right) \times \dfrac{x}{z}$

So, the numeral coefficient is $-\dfrac{8}{7}$.

Hence, option 3 is the correct option.

The degree of polynomial $8x + xy^3 − 17xz$ is:

1. 3

2. 1

3. 2

4. 4

Answer

The degree of a polynomial is the greatest of the exponents (powers) of its various terms.

Degrees of the terms are 1, (1 + 3) = 4 and (1 + 1) = 2. The greatest is 4.

Hence, option 4 is the correct option.

Which term is not a monomial:

1. $x^3y$

2. $xy$

3. $5 + y^2x$

4. $25yx^3$

Answer

An expression with only one term is a monomial. Each of $x^3y, xy$ and $25yx^3$ has one term, whereas $5 + y^2x$ has two terms, so it is a binomial and not a monomial.

Hence, option 3 is the correct option.

The constant term in the algebraic expression $4(x^2 + 1) + 5(x + 1) + 5$ is:

1. 5

2. 14

3. 9

4. 10

Answer

Solving,

$\Rightarrow 4(x^2 + 1) + 5(x + 1) + 5 \\[1em] \Rightarrow 4x^2 + 4 + 5x + 5 + 5 \\[1em] \Rightarrow 4x^2 + 5x + 14$

The term which does not contain any variable is the constant term, i.e. 14.

Hence, option 2 is the correct option.

The number of terms in the expression $5x + 5 \div x + x$ is:

1. 2

2. 4

3. 5

4. 3

Answer

Solving,

$\Rightarrow 5x + 5 \div x + x \\[1em] \Rightarrow 5x + \dfrac{5}{x} + x \\[1em] \Rightarrow (5 + 1)x + \dfrac{5}{x} \\[1em] \Rightarrow 6x + \dfrac{5}{x}$

On adding the like terms $5x$ and $x$, the expression has the two terms $6x$ and $\dfrac{5}{x}$.

Hence, option 1 is the correct option.

$3x^2 - \dfrac{2}{x} + \dfrac{5}{x^2} + 7$ is a:

1. monomial

2. multinomial

3. binomial

4. trinomial

Answer

The given expression has the four terms $3x^2, -\dfrac{2}{x}, \dfrac{5}{x^2}$ and 7. An expression with two or more than two terms is a multinomial.

Hence, option 2 is the correct option.

$4x − y + 6z − (3x − 4y + 5z)$ is equal to:

1. $7x − 5y + 11z$

2. $x + 3y + 11z$

3. $x + 3y + z$

4. $x − 3y + z$

Answer

Solving,

$\Rightarrow 4x - y + 6z - (3x - 4y + 5z) \\[1em] \Rightarrow 4x - y + 6z - 3x + 4y - 5z \\[1em] \Rightarrow (4 - 3)x + (-1 + 4)y + (6 - 5)z \\[1em] \Rightarrow x + 3y + z$

Hence, option 3 is the correct option.

$(x + y) − (y + z) + (z + x)$ is equal to:

1. $2x + y − z$

2. $2x − y + z$

3. $2x + y + z$

4. $2x$

Answer

Solving,

$\Rightarrow (x + y) - (y + z) + (z + x) \\[1em] \Rightarrow x + y - y - z + z + x \\[1em] \Rightarrow (1 + 1)x + (1 - 1)y + (-1 + 1)z \\[1em] \Rightarrow 2x$

Hence, option 4 is the correct option.

$(3x^2y) \times (50xy^2) \div (25x^2y^2)$ is equal to:

1. $6xy^2$

2. $6xy$

3. $6x^2y$

4. none of these

Answer

Solving,

$\Rightarrow (3x^2y) \times (50xy^2) \div (25x^2y^2) \\[1em] \Rightarrow \dfrac{(3 \times 50)(x^{2+1} \times y^{1+2})}{25x^2y^2} \\[1em] \Rightarrow \dfrac{150x^3y^3}{25x^2y^2} \\[1em] \Rightarrow 6x^{3-2}y^{3-2} \\[1em] \Rightarrow 6xy$

Hence, option 2 is the correct option.