Algebraic Expression and its Operations with Examples

5, 25, 100 etc.

$\mathrm{3xy}+2\frac{y}{z}+8\frac{x}{z}9$ is an algebraic expression.
So, here we have made an algebraic expression connected by + operation by combining variables such as $\mathrm{3xy},2\frac{y}{z},8\frac{x}{z}$ and a constant 9.

Consider an algebraic expression 3xy + 4yz – 8zx.
3xy, 4yz and -8zx are terms of algebraic expression.
Moreover, we can say 3xy + 4yz – 8zx has total of three terms.

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

Consider an algebraic expression 3xy + 4y + 2xy + 5z
Here, 3xy and 2xy are like terms as they have same algebraic factors x and y.

Again in algebraic expression 3xy + 4y + 2xy + 5z
4y and 5z are unlike terms as they have different algebraic factors y in term 4y and z in term 5z.

Example1: Add algebraic expressions
3xy + 7yz
8xy + 5yz
= (3xy + 7yz) + (8xy + 5yz)
Step1: Find like terms in both expressions.
So, the like terms are:
3xy and 8xy
7yz and 5yz
Step2: Add the like terms separately.
= (3xy + 8xy) + (7yz + 5yz)
= (3+8)xy + (7 + 5)yz
= 11xy + 12yz

Example2: Add algebraic expressions
3m² + 5n² + 11
7m² – 2n² + 6
= 3m² + 5n² + 11 + 7m² – 2n² + 6
Step1: Find like terms in both expressions.
So, the like terms are:
3m² and 7m²
5n² and -2n²
Step2: Add the like terms separately.
= (3m² + 7m²) + (5n² + (-2n²)) + (11 + 6)
= 10m² + (5n² – 2n²) + 17
= 10m² + 3n² + 17

Example3: Add algebraic expressions
8x² + 11y² + 6z²
9x² – 2y² – 7z²
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

Example4: Add algebraic expressions
20a + 12 and
7b + 8
= (20a + 12) + (7b + 8)
We usually combined like terms in the above examples, but we here unlike terms also exist.
Here, 20a and 7b are unlike terms.
= 20a + 7b + (12 + 8)
= 20a + 7b + 20

Example1: Subtract algebraic expressions
3xy + 7yz
8xy + 5yz
= (3xy + 7yz) – (8xy + 5yz)
Step1: Find like terms in both expressions.
So, the like terms are:
3xy and 8xy
7yz and 5yz
Step2: Subtract the like terms separately.
= (3xy – 8xy) + (7yz – 5yz)
= (3 – 8)xy + (7 – 5)yz
= -5xy + 2yz

Example2: Subtract algebraic expressions 3m² + 5n² + 11
7m² – 2n² + 6
= (3m² + 5n² + 11) – (7m² – 2n² + 6)
Step1: Find like terms in both expressions.
So, the like terms are:
3m² and 7m²
5n² and -2n²
Step2: Subtract the like terms separately.
= (3m² – 7m²) + (5n² – (-2n²)) + (11 – 6)
= -4m² + (5n² + 2n²) + 5
= -4m² + 7n² + 5

Example3: Subtract algebraic expressions
8x² + 11y² + 6z²
9x² – 2y² – 7z²
= (8x² + 11y² + 6z²) – (9x² – 2y² – 7z²)
Combine like terms.
= (8x² – 9x²) + (11y² – (-2y²)) + (6z² – (-7z²))
= -x² + (11y² + 2y²) + (6z² + 7z²)
= -x² + 13y² + 13z²

Example4: Subtract algebraic expressions 7b + 8 from 20a + 12
= (20a + 12) – (7b + 8)
here we have unlike terms.
20a and 7b are unlike terms.
= 20a – 7b + (12 – 8)
= 20a – 7b + 4

Example 1: Multiply algebraic expression 3x + 2y by 4x
= (3x + 2y) × 4x
Multiply each term of (3x + 2y) to 4x
= (3x × 4x) + (2y × 4x)
= 12x² + 8xy

Example 2: Multiply algebraic expression 4x² + 5x + 7 by 8xy
= (4x² + 5x + 7 × 8xy
Multiply each term of (4x² + 5x + 7) to 8xy
= (4x² × 8xy) + (5x × 8xy) + 7 × 8xy
= 32x³ + 40x²y + 56xy

Example 3: Multiply algebraic expression 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

Example 4: Multiply algebraic expression $\frac{3}{5}a+\frac{7}{2}b$ by $\frac{7}{2}a$
$=(\frac{3}{5}a+\frac{7}{2}b)\times \frac{7}{2}a$
Multiply each term of $(\frac{3}{5}a+\frac{7}{2}b)$ to $\frac{7}{2}a$
$=(\frac{3}{5}a\times \frac{7}{2}a)+\left(\frac{7}{2}b\times \frac{7}{2}a\right)$
$=(\frac{3}{5}\times \frac{7}{2}{a}^2)+\left(\frac{7}{2}\times \frac{7}{2}\mathrm{ab}\right)$
$=\frac{21}{10}{a}^2+\frac{49}{4}\mathrm{ab}$

Example 1: Divide 12x by 4x
$=\frac{\mathrm{12x}}{\mathrm{4x}}$
$=\frac{12\times \mathrm{x}}{4\times \mathrm{x}}$
= 3

Example 2: Divide 24x²y²z² by 4x²y
$=\frac{24x^2{y}^2{z}^2}{4x^{2}y}$
$=\frac{24\times x\times x\times y\times y\times z\times z}{4\times x\times x\times y}$
= 6yz²

Example 3: Divide 3x⁴ + 6x³ – 9x² + 12x by 3x
$=\frac{3x^4+6x^3–9x^2+12x}{3x}$
$=\frac{3x^4}{3x}+\frac{6x^3}{3x}–\frac{9x^2}{3x}+\frac{12x}{3x}$
= x³ + 2x² -3x +4

Example 4: Divide x² + 5x + 6 by x + 2
$=\frac{x^2+\mathrm{5x}+6}{x+2}$
$=\frac{x^2+\mathrm{2x}+\mathrm{3x}+6}{x+2}$       (Factorise)
$=\frac{x\mathrm{(x\; +\; 2)}+\mathrm{3(x\; +\; 3)}}{x+2}$
$=\frac{\mathrm{(x\; +\; 3)}\mathrm{(x\; +\; 2)}}{x+2}$
= x + 3