Multiplying Polynomials

Polynomials are algebraic expressions that have variables with coefficients and constants. These are of various types depending on the number of terms, such as monomial, binomial and trinomial. To multiply any two polynomials, we use the same method as we multiply two numerals.

The rules that need to be followed while multiplying any two polynomials are:

• Variables are multiplied with each other, and coefficients are multiplied with each other. For example, for multiplying 2x and 3y, 2 will be multiplied with 3 and x will be multiplied with y. Therefore, we will get 6xy.

• When the same variables are multiplied, their powers are added, for example, on multiplying x² and x³, we get x²⁺³, that is, x⁵.

Types of polynomials

Before learning about the multiplying of polynomials, let us understand the types of polynomials on the basis of the number of terms:

• Monomial: Polynomials that have only one term are known as monomials. For example, 3x, 7xy, 5, 4x².

• Binomial: Polynomials that have two terms are known as binomial. For example,

3x + 4, 4x²⁺³, 5z + 3x.

• Trinomial: Polynomials that have three terms are known as trinomials. For example, 3x + 5y + 6, x²+3x+3, 5z + 3x + 3.

Polynomial, in general, is used for expressions having more than one algebraic term.

Methods of multiplying polynomials

For the multiplication of two polynomials, there are two basic methods:

• Multiplying using the distributive property

• Multiplying using vertical method

This article aims to provide a complete understanding of these methods.

Multiplying using the distributive property

The first method of multiplying any two polynomials is using the distributive property. This law is also known as the horizontal method. The distributive property is given as:

A(B+C)=AB + BC

Here, A, B, and C are real numbers.

Taking the example of (x+4) and (x+3), let us understand the method of multiplying polynomials using the distributive law.

Step 1: The first step to multiply the polynomial is to distribute the first term of the first polynomial with the second polynomial, i.e., x(x+3).

Step 2: Now distribute the second term of the first polynomial with the second polynomial 4 (x+3).

Step 3: On distributing, we get the result as x(x+3) + 4(x+3).

Step 4: Using the distributive law in the above expression, we get: x.x +3x + 4x + 12.

Step 5: Adding the powers of the variables we get: x2 +3x + 4x + 12

Step 6: Add the like terms, and we will get the desired result: x2 +7x + 12

Multiplying using vertical method

The vertical method is the same as the normal multiplication. Let us take the same example as mentioned above (x+4)(x+3).

Step 1: Align the polynomials as shown below:

x+4

x+3

Step 2: Multiply the first polynomial (x+4) with 3;

x+4

x+3

3x+12

Step 3: Now multiply the polynomial (x+4) by x:

x+4

x+3

3x+12

x²+4x

Step 4: The last step is to add all the like terms:

 x+4

x+3

3x+12

+ x2+4x

x²+7x+12

Therefore, on multiplying (x+3) and (x+4), we get the result x2+7x+12. For both the distributive method and vertical method, the answer is the same.

Solved examples

1. Multiply (x2+y)(x+7)

Using the distributive property we get:

x²(x+7) +y(x+7)

x3+7x+xy+7

2. Multiply (y+2)(y+4) using a vertical method.

 y+2

y+4

4y +8

+ y²+2y

y²+6x+8

3. Multiply (x+3)(x²+x+2)

Using the distributive law we get:

x(x²+x+2)+3(x²+x+2)

x³+x²+2x+3×2+3x+6

x³+x²+3×2+3x+2x+6

x³+43x²+5x+6

Conclusion

Polynomials are expressions having algebraic terms such as 3x+7y+4. They are the combination of variables and constants. We can perform arithmetic operations on polynomials such as addition, subtraction, multiplication, and division.

We can do the multiplication of polynomials in two ways. One uses distributive law and the other is the vertical method. In this article, we studied both these methods in detail. Both the methods provide the same answer.