Multiplication of Algebraic Expression | Product of Algebraic Expression

# Multiplication of Algebraic Expression | Product of Algebraic Expression

Calculating Multiplication of Algebraic Expression is not that easy as you think. Students must follow some rules to find Multiplication of Algebraic Expression. If students to do any small mistakes, then it may lead to wrong answers. So, every operation is important while finding Multiplication of Algebraic Expression. Different types of problems and methods to solve problems are given in this article clearly. Students can easily understand the method of solving the Multiplication of Algebraic Expression after reading this article completely.

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## Rules to Find Multiplication of Algebraic Expression

1. Product of two same signs is positive, and also the product of different signs is negative.
2. If a is a variable and x, y are two positive integers, then (aᵐ × aⁿ) = a^(m+n)

### Types of Algebraic Expression Multiplication

There are different types of multiplication occurs while finding Algebraic Expressions
1) Multiplication of Two Monomials
2) Multiplication of a Polynomial by a Monomial
3) Multiplication of Two Binomials
4) Multiplication by Polynomial

### How to Find Multiplication of Two Monomials?

1. Write the two numbers along with the multiplication sign
2. Multiply the numbers.
3. If you find the bases are the same then add the exponents.

Product of two monomials = (Multiplication of their numerical coefficients) × (Multiplication of their variable parts)

Solved Examples

1. Find the product of 4xy and -6x²y³

Solution:
Given that 4xy and -6x²y³
4xy × -6x²y³
Multiply the coefficient. If the signs are the same, then the resultant coefficient is positive. Or else, if the signs are not the same, then the resultant coefficient is negative.
4 × -6 = -24
Multiply the variables. If the base of the variables is the same, then add the powers.
xy × x²y³
x^(1 + 2)y^(1 + 3)
x³y⁴
Multiply coefficient and variables.
-24x³y⁴

2. Find the product of 7ab², -4a²b, and -5abc?

Given that 4ab², -6a²b, and -7abc
4ab² × -6a²b × -7abc
Multiply the coefficient. If the signs are the same, then the resultant coefficient is positive. Or else, if the signs are not the same, then the resultant coefficient is negative.
4 × -6 × -7 = 168
Multiply the variables. If the base of the variables is the same, then add the powers.
ab² × a²b × abc
a^(1 + 2 + 1)b^(2 + 1 + 1)c
a⁴b⁴c
Multiply coefficient and variables.
168a⁴b⁴c

3. Find the product of 5ab and 3a³b²

Given that 5ab and 3a³b²
5ab × 3a³b²
Multiply the coefficient. If the signs are the same, then the resultant coefficient is positive. Or else, if the signs are not the same, then the resultant coefficient is negative.
5 × 3 = 15
Multiply the variables. If the base of the variables is the same, then add the powers.
ab × a³b²
a^(1 + 3)b^(1 + 2)
a⁴b³
Multiply coefficient and variables.
15a⁴b³

4. Find the product of 6x²y, 9z²x, and -6xy²z?

Given that 6x²y, 9z²x, and -6xy²z
6x²y × 9z²x × -6xy²z
Multiply the coefficient. If the signs are the same, then the resultant coefficient is positive. Or else, if the signs are not the same, then the resultant coefficient is negative.
6 × 9 × -6 = -324
Multiply the variables. If the base of the variables is the same, then add the powers.
x²y × z²x × xy²z
x^(2 + 1 + 1)y^(1 + 3)z^(2 + 1)
x⁴y⁴z³
Multiply coefficient and variables.
-324x⁴y⁴z³

### How to Find Multiplication of a Polynomial by a Monomial?

1. Use a distributive law and multiply a polynomial by a monomial.
2. Multiply each individual term in the parenthesis by a monomial.

Multiply each term of the polynomial by the monomial, using the distributive law a × (b + c) = a × b + a × c.

Solved Examples

1. 6a²b² × (2a² – 6ab + 8b²)

Solution:
Given that 6a²b² × (2a² – 6ab + 8b²)
Apply distributive law of multiplication for the given terms.
a × (b + c) = a × b + a × c.
6a²b² × (2a² – 6ab + 8b²) = (6a²b²) × (2a²) + (6a²b²) × (-6ab) + (6a²b²) × (8b²)
Find the final expression by multiplying each term.
12a⁴b² – 36a³b³ + 48a²b⁴.

The required expression is 12a⁴b² – 36a³b³ + 48a²b⁴.

2. (-9x²y) × (2x²y – 5xy² + x – 7y)

Solution:
Given that (-9x²y) × (2x²y – 5xy² + x – 7y)
Apply distributive law of multiplication for the given terms.
a × (b + c) = a × b + a × c.
(-9x²y) × (2x²y – 5xy² + x – 7y) = (-9x²y) × (2x²y) + (-9x²y) × (-5xy²) + (-9x²y) × (x) + (-9x²y) × (-7y)
Find the final expression by multiplying each term.
-18x⁴y² + 45x³y³ – 9x³y + 63x²y²

The required expression is -18x⁴y² + 45x³y³ – 9x³y + 63x²y².

3. 0(x^4 + 2x^3 + 3x^2 + 9x + 1)

Solution:
Given that 0(x^4 + 2x^3 + 3x^2 + 9x + 1)
Any polynomial multiply by zero is zero.
Therefore, multiplying (x^4 + 2x^3 + 3x^2 + 9x + 1) with 0 is 0.

4. 1 (5 x^4 – 8 )

Solution:
Given that 1 (5 x^4 – 8 )
Any polynomial multiply by 1 is the polynomial itself.
Therefore, multiplying (5 x^4 – 8 ) with 1 is (5 x^4 – 8).

The required expression is (5 x^4 – 8).

### 3. How to Find Multiplication of Two Binomials?

Use two methods to find the Multiplication of Two Binomials. Students can use a horizontal method or Column wise multiplication to find Multiplication of Two Binomials.

How to Find Multiplication of Two Binomials using Horizontal method?

1. Firstly, note down two binomials.
2. Apply the distributive law of multiplication over addition twice.
3. Find the final expression of multiplication.

How to Find Multiplication of Two Binomials using Column wise multiplication?

1. Write one binomial expression under another expression.
2. Multiply the first binomial expression with the first term of the second binomial expression.
3. Multiply the first binomial expression with the second term of the second binomial expression.
4. Note down the first resultant expression and write the second resultant expression below the first resultant expression with like terms comes at the same column.
5. Add the first and second expressions to get the final expression.

Solved Examples

1. Multiply (m + n) × (r + s)

Solution:

Horizontal Method:
Note down two binomials.
(m + n) × (r + s)
Apply the distributive law of multiplication over addition twice.
m × (r + s) + n × (r + s)
(m × r + m × s) + (n × r + n × s)
mr + ms + nr + ns

The required expression is mr + ms + nr + ns.

Column wise multiplication

Write one binomial expression under another expression
m + n
× (r + s)
—————————-
rm + rn —> Multiplication of r with (m + n)
+ ms + ns —> Multiplication of s with (m + n)
—————————-
mr + ms + nr + ns

The required expression is mr + ms + nr + ns.

2. Multiply (2x + 4y) and (4x – 6y)

Solution:

Horizontal Method:
Note down two binomials.
(2x + 4y) and (4x – 6y)
Apply the distributive law of multiplication over addition twice.
2x × (4x – 6y) + 4y × (4x – 6y)
(2x × 4x – 2x × 6y) + (4y × 4x – 4y × 6y)
(8x² – 12xy) + (16xy – 24y²)
8x² – 12xy + 16xy – 24y²
8x² + 4xy – 24y².

The required expression is 8x² + 4xy – 24y².

Column wise multiplication
Write one binomial expression under another expression
(2x + 4y)
× (4x – 6y)
—————————-
8x² + 16xy —> Multiplication of 4x with (4x – 6y)
– 12xy – 24y² —> Multiplication of 4y with (4x – 6y)
—————————-
8x² + 4xy – 24y².

The required expression is 8x² + 4xy – 24y².

3. Multiply (4x² + 2y²) by (3x² + 5y²)

Solution:

Horizontal Method:
Note down two binomials.
(4x² + 2y²) × (3x² + 5y²)
Apply the distributive law of multiplication over addition twice.
4x² (3x² + 5y²) + 2y² (3x² + 5y²)
(12x⁴ + 20x²y²) + (6x²y² + 10y⁴)
12x⁴ + 20x²y² + 6x²y² + 10y⁴
12x⁴ + 26x²y² + 10y⁴

The required expression is 12x⁴ + 26x²y² + 10y⁴

Column wise multiplication
Write one binomial expression under another expression
(4x² + 2y²)
× (3x² + 5y²)
—————————-
12x⁴ + 20x²y² —> Multiplication of 3x² with (4x² + 2y²)
6x²y² + 10y⁴ —> Multiplication of 5y² with (4x² + 2y²)
—————————-
12x⁴ + 26x²y² + 10y⁴

The required expression is 12x⁴ + 26x²y² + 10y⁴

### 4. How to Find Multiplication by Polynomial?

Apply horizontal method or column multiplication to find Multiplication by Polynomial.

Solved Examples:

1. Multiply (6x² – 4x + 9) by (3x – 7)

Solution:

Horizontal Method:
Given that (6x² – 4x + 9) by (3x – 7)
(3x – 7) × (6x² – 4x + 9)
3x (6x² – 4x + 9) – 7 (6x² – 4x + 9)
(18x³ – 12x² + 27x ) + (- 42x² + 28x – 63)
18x³ – 12x² + 27x – 42x² + 28x – 63
18x³ – 54x² + 55x – 63

The required expression is 18x³ – 54x² + 55x – 63

Column wise multiplication
Write one binomial expression under another expression
(6x² – 4x + 9)
× (3x – 7)
—————————-
18x³ – 12x² + 27x —> Multiplication of 3x with (6x² – 4x + 9)
– 42x² + 28x – 63 —> Multiplication of -7 with (6x² – 4x + 9)
—————————-
18x³ – 54x² + 55x – 63

The required expression is 18x³ – 54x² + 55x – 63

2. Multiply (3x² – 6x + 2) by (2x² + 9x – 5)

Solution:

Horizontal Method:
Given that (3x² – 6x + 2) by (2x² + 9x – 5)
2x² (3x² – 6x + 2) + 9x (3x² – 6x + 2) – 5 (3x² – 6x + 2)
(6x⁴ – 12x³ + 4x²) + ( + 27x³ – 54x² + 18x ) + (- 15x² + 30x – 10 )
6x⁴ – 12x³ + 4x² + 27x³ – 54x² + 18x – 15x² + 30x – 10
6x⁴ + 15x³ – 65x² + 48x – 10

The required expression is 6x⁴ + 15x³ – 65x² + 48x – 10

Column wise multiplication
Write one binomial expression under another expression
(3x² – 6x + 2)
× (2x² + 9x – 5)
—————————-
6x⁴ – 12x³ + 4x² —> Multiplication of 2x² with (6x² – 4x + 9)
+ 27x³ – 54x² + 18x —> Multiplication of 9x with (6x² – 4x + 9)
– 15x² + 30x – 10 —> Multiplication of – 5 with (6x² – 4x + 9)
—————————-
6x⁴ + 15x³ – 65x² + 48x – 10

The required expression is 6x⁴ + 15x³ – 65x² + 48x – 10

3. Multiply (3x³ – 4x² – 2x + 9) by (5 – 6x + 7x²)

Solution:

Horizontal Method:
Given that (3x³ – 4x² – 2x + 9) by (5 – 6x + 7x²)
5 (3x³ – 4x² – 2x + 9) – 6x (3x³ – 4x² – 2x + 9) + 7x² (3x³ – 4x² – 2x + 9)
(15x³ – 20x² – 10x + 45) + (-18x⁴ + 24x³ + 12x² – 54x ) + (21x⁵ – 28x⁴ – 14x³ + 63x²)
15x³ – 20x² – 10x + 45 -18x⁴ + 24x³ + 12x² – 54x + 21x⁵ – 28x⁴ – 14x³ + 63x²
21x⁵ -46x⁴ + 25x³ + 55x² – 10x + 45

The required expression is 21x⁵ -46x⁴ + 25x³ + 55x² – 10x + 45

Column wise multiplication
Write one binomial expression under another expression
(3x³ – 4x² – 2x + 9)
× (5 – 6x + 7x²)
—————————-
15x³ – 20x² – 10x + 45 —> Multiplication of 5 with (3x³ – 4x² – 2x + 9)
-18x⁴ + 24x³ + 12x² – 54x —> Multiplication of – 6x with (3x³ – 4x² – 2x + 9)
21x⁵ – 28x⁴ – 14x³ + 63x² —> Multiplication of 7x² with (3x³ – 4x² – 2x + 9)
—————————-
21x⁵ -46x⁴ + 25x³ + 55x² – 10x + 45

The required expression is 21x⁵ -46x⁴ + 25x³ + 55x² – 10x + 45

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