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 × 10n^{2}= m × 10 × n × n = 10mn^{2}

(ii)20t × 3n = 20 × t × 3 × n = 60tn

(iii)100q × (-8qzr) = 100 × q × (-8) × q × z × r= -800q^{2}zr

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 = 400pq

^{2}rNow multiply this product with 8pr

Final product is 400pq

^{2}r × 8pr = 3200p^{2}q^{2}r^{2}We can obtain the same solution by first multiplying the coefficients 100 × 4 × 8 = 3200

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

^{2}q^{2}r^{2}So, the final product is 3200p

^{2}q^{2}r^{2}

**Question 2. Find 5pqr × 10 rst**

**Solution:**

Multiply the coefficients 5 × 10 =50

Multiply the algebraic coefficients = pqr × rst = pqr

^{2}stSo, Product = 50pqr

^{2}stThe 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 + 95p

^{2}q

### 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)]

[6x

^{2}– 10xy] – [12xy – 20y^{2}]6x

^{2}– 10xy – 12xy – 20y^{2}6x

^{2}– 20y^{2 }– 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)]

[4x

^{2}+ 2xy] + [8xy + 4y^{2}]4x

^{2}+ 2xy + 8xy + 4y^{2}4x

^{2}+ 4y^{2 }+ 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) = 12m

^{2}– 15mSo, 3m (4m – 5) + 4 = 12m

^{2}– 15m + 4Now 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)

= z

^{2 }– 4z + 4z – 16= z

^{2}– 16

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

**Solution: **

Given: (m – n) × (3m + 5n)

Using distributed law

= m(3m + 5n) – n(3m + 5n)

= 3m

^{2}+ 5mn – 3mn – 5n^{2}= 3m

^{2 }+ 2mn – 5n^{2}

**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)

= 2m

^{2}+ 3mn + mr – 2mn – 3n^{2}– nr= 2m

^{2}+ mn – 3n^{2 }+ 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)

= p

^{2}+ pq + pr + pq + q^{2}+ qr= p

^{2}+ q^{2}+ 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 t

^{2}+ 5tq= 15t

^{2 }+ 37t + 5tq + 4q + 20