Fundamental Concepts of Algebra — Multiple Choice Questions

Out of the algebraic expressions, $8 - 5x, x^2 - \dfrac{3}{5}x + \dfrac{12}{x^2}, 8 - \dfrac{5}{x}$ and 8.7, the polynomials are:

1. $8 - 5x, 8 - \dfrac{5}{x}$ and 8.7

2. $8 - x$ and $x^2 - \dfrac{3}{5}x + \dfrac{12}{x^2}$

3. $8 - 5x$ and 8.7

4. $x^2 - \dfrac{3}{5}x + \dfrac{12}{x^2} \text{ and } 8 - \dfrac{5}{x}$

Answer

$\dfrac{12}{x^2} = 12x^{-2} \text{ and } \dfrac{5}{x} = 5x^{-1}$ have negative exponents of $x$, so those two expressions are not polynomials. In $8 - 5x$ and 8.7, the exponents of $x$ are whole numbers.

Hence, option 3 is the correct option.

The value of $3y(x^3 - z)$ for $x = 2, y = 3$ and $z = 5$ is:

1. 117

2. 27

3. 36

4. none of these

Answer

Substituting $x = 2, y = 3$ and $z = 5$,

$\Rightarrow 3y(x^3 - z) \\[1em] \Rightarrow 3(3)[(2)^3 - 5] \\[1em] \Rightarrow 9(8 - 5) = 9 \times 3 = 27.$

Hence, option 2 is the correct option.

The sum of $4x + 8x^2 + 1, 13 - x^2$ and $6x - 10 - 8x^2$ is:

1. $-x^2 + 10x + 4$

2. $x^2 - 10x - 4$

3. $-x^2 - 10x + 4$

4. $x^2 + 10x - 4$

Answer

Solving,

$\Rightarrow (4x + 8x^2 + 1) + (13 - x^2) + (6x - 10 - 8x^2)\\[1em] \Rightarrow (8 - 1 - 8)x^2 + (4 + 6)x + (1 + 13 - 10)\\[1em] \Rightarrow -x^2 + 10x + 4.$

Hence, option 1 is the correct option.

$(3x^2 + 7 - 6x) - (3x - 4x^2 - 15)$ is equal to:

1. $-x^2 - 3x - 5$

2. $7x^2 - 9x + 22$

3. $x^2 + 3x + 5$

4. $7x^2 - 9x - 22$

Answer

Solving,

$\Rightarrow (3x^2 + 7 - 6x) - (3x - 4x^2 - 15)\\[1em] \Rightarrow 3x^2 + 7 - 6x - 3x + 4x^2 + 15\\[1em] \Rightarrow 7x^2 - 9x + 22.$

Hence, option 2 is the correct option.

$(5x^3y^2z^5) \times (-4x^2y^3)$ is equal to:

1. $20x^5y^5z^5$

2. $-20x^3yz^5$

3. $-20x^5y^5z^5$

4. $-9x^3y^3z^5$

Answer

Solving,

$\Rightarrow (5x^3y^2z^5) \times (-4x^2y^3)\\[1em] \Rightarrow (5 \times -4) \times x^{3+2} \times y^{2+3} \times z^5\\[1em] \Rightarrow -20x^5y^5z^5.$

Hence, option 3 is the correct option.

$(3x - 4y) \times (5x - 6y)$ is equal to:

1. $15x^2 + 2xy + 24y^2$

2. $15x^2 - 2xy + 24y^2$

3. $15x^2 + 30xy + 24y^2$

4. $15x^2 - 38xy + 24y^2$

Answer

Solving,

$\Rightarrow (3x - 4y)(5x - 6y)\\[1em] \Rightarrow 15x^2 - 18xy - 20xy + 24y^2\\[1em] \Rightarrow 15x^2 - 38xy + 24y^2.$

Hence, option 4 is the correct option.

$(x - 1) (x^2 + x + 1)$ is equal to:

1. $x^3 - 1$

2. $x^3 + 1$

3. $x^3 - x + 1$

4. $x^3 + x + 1$

Answer

Solving,

$\Rightarrow (x - 1)(x^2 + x + 1)\\[1em] \Rightarrow x^3 + x^2 + x - x^2 - x - 1\\[1em] \Rightarrow x^3 - 1.$

Hence, option 1 is the correct option.

$(x + 1) (x^2 - x + 1)$ is equal to:

1. $x^3 - 1$

2. $x^3 + 1$

3. $x^3 - x + 1$

4. $x^3 + x + 1$

Answer

Solving,

$\Rightarrow (x + 1)(x^2 - x + 1)\\[1em] \Rightarrow x^3 - x^2 + x + x^2 - x + 1\\[1em] \Rightarrow x^3 + 1.$

Hence, option 2 is the correct option.

$(-63x^5y^3z^{10}) \div (-7x^2y^5z^7)$ is equal to:

1. $-9x^3z \div y^2$

2. $9x^3y^2z$

3. $9x^3z^3 \div y^2$

4. $-9x^3y^2z$

Answer

Solving,

$\Rightarrow \dfrac{-63x^5y^3z^{10}}{-7x^2y^5z^7} \\[1em] \Rightarrow 9 \times x^{5-2} \times y^{3-5} \times z^{10-7} \\[1em] \Rightarrow \dfrac{9x^3z^3}{y^2}$

Hence, option 3 is the correct option.

$(4x^3y - 12x^2y^2) \div (-4xy)$ is equal to:

1. $4x^2 + 3xy$

2. $4x^2 - 3x$

3. $-x^2 + 3xy$

4. $-x^2 - 3xy$

Answer

Solving,

$\Rightarrow \dfrac{4x^3y - 12x^2y^2}{-4xy} \\[1em] \Rightarrow \dfrac{4x^3y}{-4xy} - \dfrac{12x^2y^2}{-4xy} \\[1em] \Rightarrow -x^2 + 3xy$

Hence, option 3 is the correct option.

$2x - 2(x + 3) + 4(x - 1)$ is equal to:

1. $-4x - 10$

2. $-4x + 10$

3. $4x + 10$

4. $4x - 10$

Answer

Solving,

$\Rightarrow 2x - 2(x + 3) + 4(x - 1)\\[1em] \Rightarrow 2x - 2x - 6 + 4x - 4\\[1em] \Rightarrow 4x - 10.$

Hence, option 4 is the correct option.

The degree of $(8 - x) \times (3 - 2x)$ is:

1. 2

2. $2x^2$

3. $-19x$

4. none of these

Answer

Solving,

$\Rightarrow (8 - x)(3 - 2x) \\[1em] \Rightarrow 24 - 16x - 3x + 2x^2 \\[1em] \Rightarrow 2x^2 - 19x + 24.$

The highest power of $x$ is 2, so the degree is 2.

Hence, option 1 is the correct option.

In expression $3xy^2 - 5yz + 2yz^2$, the coefficient of $x$ is:

1. $3y^2 - 5yz + 2yz^2$

2. $y^2 - 2yz^2$

3. $3y^2$

4. $3y^2 - 5yz$

Answer

Only the term $3xy^2$ contains $x$, and $3xy^2 = (3y^2) \times x$. So the coefficient of $x$ is $3y^2$.

Hence, option 3 is the correct option.

$(9 + x + 2x^2) - (3x^2 - 5x) - (3 + 2x - x^2)$ is equal to:

1. $6 + 4x$

2. $6 - 4x$

3. $x^2 + 4x + 6$

4. $x^2 - 4x - 6$

Answer

Solving,

$\Rightarrow (9 + x + 2x^2) - (3x^2 - 5x) - (3 + 2x - x^2)\\[1em] \Rightarrow 9 + x + 2x^2 - 3x^2 + 5x - 3 - 2x + x^2\\[1em] \Rightarrow (2 - 3 + 1)x^2 + (1 + 5 - 2)x + (9 - 3)\\[1em] \Rightarrow 6 + 4x.$

Hence, option 1 is the correct option.

For $x = 2$, the value of the expression $3(4x - 1) - 2(5 - 2x)$ is:

1. -19

2. 19

3. 10

4. -10

Answer

Substituting $x = 2$,

$3(4x - 1) - 2(5 - 2x) = 3(8 - 1) - 2(5 - 4) = 3(7) - 2(1) = 21 - 2 = 19$.

Hence, option 2 is the correct option.