Coefficients And Terms of Algebraic Expressions With Basic Operations

5, 25, 100 etc.

$\mathrm{3xy}+\frac{\mathrm{2y}}{z}+\frac{\mathrm{8x}}{z}+9$

This example is of an algebraic expression connected by + operation by combining variables $\mathrm{3xy},\frac{\mathrm{2y}}{z},\frac{\mathrm{8x}}{z}$ and a constant 9.

3xy + 4yz - 8zx

3xy, 4yz and -8zx are terms

Moreover, we can say 3xy + 4yz - 8zx has a total of three terms.

In 3xy + 4yz - 8zx, 3 is numerical factor and xy is an algebraic factor.

4 is a numerical factor and yz is an algebraic factor.

-8 is a numerical factor and zx is an algebraic factor.

3xy + 4y + 2xy + 5z

Here, 3xy and 2xy are like terms as they have the same algebraic factors x and y.

3xy + 4y + 2xy + 5z

Here, 4y and 5z are unlike terms as they have different algebraic factors y in term 4y and z in term 5z.

Add 3xy + 7yz and 8xy + 5yz

= (3xy + 7yz) + (8xy + 5yz)

Step 1: Find like terms in both expressions.

So, the like terms are:

3xy and 8xy

7yz and 5yz

Step 2: Add the like terms separately.

= (3xy + 8xy) + (7yz + 5yz)

= (3+8)xy + (7 + 5)yz

= 11xy + 12yz

Add 3m² + 5n² + 11 and 7m² - 2n² + 6

= 3m² + 5n² + 11 + 7m² - 2n² + 6

Step 1: Find like terms in both expressions.

So, the like terms are:

3m² and 7m²

5n² and -2n²

Step 2: Add the like terms separately.

= (3m² + 7m²) + (5n² + (-2n²)) + (11 + 6)

= 10m² + (5n² - 2n²) + 17

= 10m² + 3n² + 17

Add 8x² + 11y² + 6z², 9x² - 2y² - 7z² and 2x² + 4y² + 5z²

= (8x² + 11y² + 6z²) + (9x² - 2y² - 7z²) + (2x² + 4y² + 5z²)

Combine like terms.

= (8x² + 9x² + 2x²) + (11y² - 2y² + 4y²) + (6z² - 7z² + 5z²)

= 19x² + 13y² + 4z²

Add 20a + 12 and 7b + 8

= (20a + 12) + (7b + 8)

Here, 20a and 7b are unlike terms.

= 20a + 7b + (12 + 8)

= 20a + 7b + 20

Subtract (8xy + 5yz) from (3xy + 7yz)

= (3xy + 7yz) - (8xy + 5yz)

Step 1: Find like terms in both expressions.

So, the like terms are:

3xy, 8xy

and 7yz, 5yz

Step 2: Subtract the like terms separately.

= (3xy - 8xy) + (7yz - 5yz)

= (3 - 8)xy + (7 - 5)yz

= -5xy + 2yz

Subtract 7m² - 2n² + 6 from 3m² + 5n² + 11

= (3m² + 5n² + 11) - (7m² - 2n² + 6)

Step 1: Find like terms in both expressions.

So, the like terms are:

3m² and 7m²

5n² and -2n²

Step 2: Subtract the like terms separately.

= (3m² - 7m²) + (5n² - (-2n²)) + (11 - 6)

= -4m² + (5n² + 2n²) + 5

= -4m² + 7n² + 5

Subtract 9x² - 2y² - 7z² from 8x² + 11y² + 6z²

= (8x² + 11y² + 6z²) - (9x² - 2y² - 7z²)

= (8x² - 9x²) + (11y² - (-2y²)) + (6z² - (-7z²))

= -x² + (11y² + 2y²) + (6z² + 7z²)

= -x² + 13y² + 13z²

Subtract 7b + 8 from 20a + 12

= (20a + 12) - (7b + 8)

20a and 7b are unlike terms.

= 20a - 7b + (12 - 8)

= 20a - 7b + 4

Multiply 3x + 2y by 4x

= (3x + 2y) × 4x

Multiply each term of (3x + 2y) to 4x

= (3x × 4x) + (2y × 4x)

= 12x² + 8xy

Multiply 4x² + 5x + 7 by 8xy

= (4x² + 5x + 7 × 8xy

Multiply each term of (4x² + 5x + 7) to 8xy

= (4x² × 8xy) + (5x × 8xy) + 7 × 8xy

= 32x³y + 40x²y + 56xy

Multiply 7x - 3y by 2y + 5z

= (7x - 3y) × (2y + 5z)

Multiply each term of (7x - 3y) to (2y + 5z)

= 7x × (2y + 5z) - 3y × (2y + 5z)

Again, multiply each term of (2y + 5z) with 7x

and multiply each term of (2y + 5z) with 3y

= 7x × 2y + 7x × 5z - 3y × 2y - 3y × 5z

= 14xy + 35xz - 6y² - 15yz

Multiply $\frac{\mathrm{3a}}{5}+\frac{\mathrm{7b}}{2}$ by $\frac{\mathrm{7a}}{2}$

Multiply each term of $(\frac{\mathrm{3a}}{5}+\frac{\mathrm{7b}}{2})$ to $\frac{\mathrm{7a}}{2}$

$=(\frac{\mathrm{3a}}{5}\times \frac{\mathrm{7a}}{2})+(\frac{\mathrm{7b}}{2}\times \frac{\mathrm{7a}}{2})$

$=(\frac{3}{5}\times \frac{7}{2}{a}^2)+(\frac{7}{2}\times \frac{7}{2}\mathrm{ab})$

$=\frac{{\mathrm{21a}}^2}{10}+\frac{\mathrm{49ab}}{4}$

Divide 12x by 4x

$=\frac{\mathrm{12x}}{\mathrm{4x}}$

$=\frac{\mathrm{12\; \times \; x}}{\mathrm{4\; \times \; x}}$

= 3

Divide 24x²y²z² by 4x²y

$=\frac{\mathrm{12\; \times \; x}}{\mathrm{4\; \times \; x}}$

$=\frac{\mathrm{24\; \times \; x\; \times \; x\; \times \; y\; \times \; y\; \times \; z\; \times \; z}}{\mathrm{4\; \times \; x\; \times \; x\; \times \; yx}}$

= 6yz²

Divide 3x⁴ + 6x³ - 9x² + 12x by 3x

$=\frac{{\mathrm{3x}}^4+{\mathrm{6x}}^3-{\mathrm{9x}}^2+\mathrm{12x}}{\mathrm{3x}}$

$=\frac{{\mathrm{3x}}^4}{\mathrm{3x}}+\frac{{\mathrm{6x}}^3}{\mathrm{3x}}-\frac{{\mathrm{9x}}^2}{\mathrm{3x}}+\frac{\mathrm{12x}}{\mathrm{3x}}$

= x³ + 2x² - 3x + 4

Divide x² + 5x + 6 by x + 2

$=\frac{x^2+\mathrm{5x}+6}{\mathrm{x\; +\; 2}}$

$=\frac{x^2+\mathrm{2x}+\mathrm{3x}+6}{\mathrm{x\; +\; 2}}$    (Factorise)

$=\frac{x(x+2)+3(x+2)}{\mathrm{x\; +\; 2}}$

$=\frac{\mathrm{(x\; +\; 3)(x\; +\; 2)}}{\mathrm{x\; +\; 2}}$

= x + 3