Multiplying Polynomials

Polynomial is an algebraic expression consisting of variable and coefficient. Variable is also at times called indeterminate. We can perform any of the operations using polynomials whether it be multiplication, division, subtraction, or addition. In this article, we are going to learn how to multiply polynomials.

Multiplying Monomial by a Monomial

In mathematic, a monomial is an expression in algebra that contains only one variable, it can be a number, whole number, and a variable that multiplies together like 2x, 4mn, etc. Now, we will learn how to multiply a monomial by a monomial:

Multiplying two monomials:

As we know that:

3 × m = m + m + m  

Similarly, 3 × (10m) = 10m +10m +10m = 30 m  

Examples:

(i) m × 10n² = m × 10 × n × n = 10mn²

(ii) 20t × 3n = 20 × t × 3 × n = 60tn

(iii) 100q × (-8qzr) = 100 × q × (-8) × q × z × r= -800q²zr

As we can see from these examples that the coefficient of product is equal to the product of coefficients of the first and second monomial. 

Multiplying three or more monomials:

To find the product of three or more monomials, we can first multiply any two monomials and then multiply this product with the remaining monomials. We can extend this method to find the product of any number of monomials.

Examples

Question 1. Evaluate 100pq × 4qr × 8pr  

Solution:

Given: 100pq × 4qr × 8pr  

So, we shall first multiply 100 pq and 4qr = 400pq²r

Now multiply this product with 8pr  

Final product is 400pq²r × 8pr = 3200p²q²r²

We can obtain the same solution by first multiplying the coefficients 100 × 4 × 8 = 3200

The product of algebraic coefficients is pq × qr × pr = p²q²r²

So, the final product is 3200p²q²r²

Question 2. Find 5pqr × 10 rst

Solution:

Multiply the coefficients 5 × 10 =50

Multiply the algebraic coefficients = pqr × rst = pqr²st

So, Product = 50pqr²st

The result of multiplication doesn’t depend on the order in which multiplication is carried out.

Multiplying Monomial by a Polynomial

We are allowed to multiply a monomial by a polynomial using the following steps:

Step 1: Arrange the monomial and polynomial in a line.

Step 2: Now use distributed law to separate them. 

Step 3: After separation multiply the first term with the second term

Step 4: simplifies the result(if needed).

Examples

Question 1.  Multiply 20m × (10n + 3).

Solution:

Given: 20m x (10n + 3) 

Using the distributive laws,

= (20m × 10n) + (20m × 3)

= 200mn + 60m  

Question 2.  Find the product 19p × (2q + 3z + 5pq)  

Solution:

Given: 19p × (2q + 3z + 5pq)  

Using the distributive law,

= (19p × 2q) + (19p × 3z) + (19p × 5pq)

= 38pq + 57pz + 95p²q

Multiplying Polynomial 

We are allowed to multiply one polynomial with another polynomial using the following steps:

Step 1: Arrange the polynomials in a line.

Step 2: Now use distributed law to separate them. 

Step 3: After separation multiply the first term with the second term

Step 4: simplifies the result(if needed).

Using these steps you can multiply multiple polynomials with each other. And when the two polynomial multiplies then the degree of the resulting polynomial is always higher.

Examples:

Question 1. Multiply (2x – 4y) and (3x – 5y).

Solution:

Given: (2x – 4y) × (3x – 5y)

Using the distributive laws,

[2x × (3x – 5y)] – [4y × (3x – 5y)]

[6x² – 10xy] – [12xy – 20y²]

6x² – 10xy – 12xy – 20y²

6x² – 20y² – 22xy 

Question 2. Multiply (2x + 4y) and (2x + y).

Solution:

Given: (2x + 4y) × (2x + y)

Using the distributive laws,

[2x × (2x + y)] + [4y × (2x + y)]

[4x² + 2xy] + [8xy + 4y²]

4x² + 2xy + 8xy + 4y²

4x² + 4y² + 10xy 

Question 3. Find the value of 3m (4m – 5) + 4 when m = 1

Solution

Given: 3m (4m – 5) + 4, m = 1

3m(4m – 5) = 12m² – 15m 

So, 3m (4m – 5) + 4 = 12m² – 15m + 4

Now put the value m = 1

= 12(1)² – 15 (1) + 4 

= 12 – 15 + 4

= 1

Types of polynomial multiplication:

There are two types of polynomial multiplication are available: 

1. Multiplying binomial by a binomial

We are allowed to multiply one binomial with another binomial using the following steps:

Step 1: Arrange the binomials in a line.

Step 2: Now use distributed law to separate them. 

Step 3: After separation multiply the first term with the second term

Step 4: Combine similar terms(if available).

Examples:

Question 1. Multiply (t – 5) and (3m + 5)

Solution: 

Given: (t – 5) × (3m + 5)

Using distributed law

t(3m + 5) – 5(3m + 5)

3tm + 5t – 15m – 25

Question 2. Multiply (z + 4) and (z – 4)

Solution: 

Given: (z + 4) × (z – 4)

Using distributed law

= z(z – 4) + 4(z – 4)

= z² – 4z + 4z – 16

= z² – 16

Question 3. Multiply (m – n) and (3m + 5n)

Solution: 

Given: (m – n) × (3m + 5n)

Using distributed law

= m(3m + 5n) – n(3m + 5n)

= 3m² + 5mn – 3mn – 5n²

= 3m² + 2mn – 5n²

2. Multiplying binomial and a trinomial

We are allowed to multiply one binomial with another trinomial using the following steps:

Step 1: Arrange the binomial and trinomial in a line.

Step 2: Now use distributed law to separate them. 

Step 3: After separation, each of two terms of the binomial gets multiplied by each of three terms of the trinomial.

Step 4: Combine similar terms(if available).

Examples

Question 1. Simplify (m – n)(2m + 3n + r) 

Solution:

Given: (m – n)(2m + 3n + r) 

Using distributed law

= m(2m + 3n + r) – n(2m + 3n + r) 

= 2m² + 3mn + mr – 2mn – 3n² – nr

= 2m² + mn – 3n² + mr – nr

Question 2. Evaluate (p + q) (p + q + r)

Solution:

Given: (p + q)(p + q + r)

Using distributed law

= p(p + q + r) + q(p + q + r)

= p² + pq + pr + pq + q² + qr

= p² + q² + 2pq + pr + qr

Question 3. Evaluate (4 + 5t)(5 + 3t + q)

Solution

Given: (4 + 5t)(5 + 3t + q)

Using distributed law

= 4(5 + 3t + q) + 5t (5 + 3t + q)

= 20 + 12t + 4q + 25t + 15 t² + 5tq

= 15t² + 37t + 5tq + 4q + 20