Multiplying Polynomials

Polynomial multiplication is the process of multiplying each term of one polynomial by every term of another polynomial.

Steps to Multiply Polynomials

Multiply each term of one polynomial with each term in the other polynomial.
Add exponents of same variables
Combine like terms

Types of Polynomial Multiplication

There are different types of polynomial multiplication based on the number of terms:

1. Multiplying Polynomials with Like Bases

To multiply the polynomial in which we have the same variables, we multiply the polynomials using the exponent rules:

Step 1: Multiply the coefficients of both the variables.
Step 2: To multiply the variables, we use the laws of exponents.

For example, multiply the polynomials 3x⁵ and 5x.²

(3x⁵)(5x²) = (3 · 5)(x⁵ · x²) = 15(x⁵⁺²) = 15x⁷

2. Multiplying Polynomials with Different Variables

Polynomials with different variables are multiplied together by

Step 1: Multiply the coefficient of both the variables.
Step 2: Use the laws of exponents to multiply the polynomial or write different variables together to get the required product of the variable.

For example, multiply the polynomials 3x⁵ and 5y.²

(3x⁵)(5y²) = (3 · 5)(x⁵ · y²) = 15x⁵y ²

3. Multiplying a Monomial by a Polynomial

To multiply a polynomial and a monomial, we need to multiply each and every term of the polynomial with the monomial.

For example, the product of 5x and (5x² + 2x + 6)

5x (5x²+ 2x + 6) = (5x × 5x²) + (5x × 2x) + (5x × 6) = 25x³+ 10x²+ 30x

4. Multiplying a Polynomial by a Polynomial

To multiply two polynomials, we need to multiply each and every term of one polynomial with each and every term of the other polynomial.

For example, the product of (5x² + 2x + 6) and (x² + 2x + 3)

(5x² + 2x + 6) × (1x² + 2x + 3)
= (5x² × 1x²) + (5x² × 2x) + (5x² × 3) + (2x × 1x²) + (2x × 2x) + (2x × 3) + (6 × 1x²) + (6 × 2x) + (6 × 3)
= 5x⁴ +10x³ + 15x² + 2x³ + 4x² + 6x + 6x² + 12x + 18
= 5x⁴ +12x³ + 21x² + 18x + 18

5. Multiplying a Monomial by a Monomial

Two monomials are easily multiplied. We should follow the following steps to multiply the monomial.

Step 1: Multiply the coefficients of both polynomials together to get the coefficient of the resultant polynomial.
Step 2: Multiply the variables of both the polynomials to get the required product.

For example, multiply the polynomials 20x⁵y and 3xy.²

(20x⁵y)(3xy²) = (20 · 3)(x⁵y · xy²) = 60(x⁵⁺¹)(y¹⁺²) = 60x⁶y³

6. Multiplying a Binomial by a Binomial

Two binomials can be multiplied using the distributive properties. The distributive property of the algebra is (a + b) · (c + d) = (a · c) + (a · d) +( b · c) +( b · d)

Using this property we can easily multiply two binomials.

Step 1: Write the polynomials in the form (a ± b). (c ± d)
Step 2: Use the distributive property.
Step 3: Simplify to get the required product.

For example, multiply (3x + 4y)(5x² + 2xy)

(3x + 4y)(5x² + 2xy) = (3x)(5x²) + (3x)(2xy) + (4y)(5x²) + (4y)(2xy) = 15x³ + 6x²y + 20x²y + 8xy² = 15x³ + 26x²y + 8xy²

7. Multiplying Binomials by Box Method

The box method (or area model) is a visual way to multiply polynomials, particularly binomials. It helps organize calculations and makes it easy to combine like terms.

Steps to Multiply Binomials using the Box Method

Draw a 2 × 2 box and write one binomial on the top and the other on the side.
Multiply each row term with each column term and fill the boxes with the products.
Combine like terms.

For example, multiply (2x + 2) (x + 2x)

\begin{array}{|c|c|c|}\hline \times & 2x^{2} & + 2\\\hline x & 2x^{3} & 2x \\ \hline +2x & 4x^{3} & 4x \\ \hline \end{array}
Combine like terms: 2x³ + 4x³ + 2x + 4x = 6x³ + 6x
Thus, (2x²+ 2) (x + 2x) = 6x³ + 6x

Types of Polynomials
Zeros of Polynomial
Algebraic Identities

Solved Examples

Example 1: Find the product of (3x² + 1x + 2) and (1x² + 2x + 1)

(3x² + 1x + 2) × (1x² + 2x + 1)
(3x² × 1x²) + (3x² × 2x) + (3x² × 1) + (1x × 1x²) + (1x × 2x) + (1x × 1) + (2 × 1x²) + (2 × 2x) + (2 * 1)
3x⁴ + 6x³ + 3x² + 1x³ + 2x² + 1x + 2x² + 4x + 2
3x⁴ + 7x³ + 7x² + 5x + 2

Example 2: Find the product of (5xy + 1) and (2z + 3)

(5xy + 1) × (2z + 3)
(5xy × 2z) + (5xy × 3) + (1 × 2z) + (1 × 3)
10xyz + 15xy + 2z + 3

Example 3: Find the Product of (3xyz) and (2x + 6)

(3xyz) × (2x + 6)
(3xyz × 2x) + (3xyz × 6)
6x²yz +18xyz

Example 4: Find the product of (−a³b) and (2ab³)

(−a³b) × (2ab³) = -2a⁴b⁴

Example 5: Find the product of (xy + 2y) and (a + b)

(xy + 2y) × (a + b)
(xy × a) + (xy × b) + (2y × a) + (2y × b)
axy + bxy + 2ay + 2by

Practice Questions

Question 1: Multiply 2x² and 3xy.

Question 2: Multiply (3x² – 5y) and (4x – y).

Question 3: Multiply (x + 2y) and (3x² − 4xy + 5).

Question 4: Multiply (xy – 3) and (2x² – 9y).