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Multiplying Polynomials – Explanation & Examples

Many students will find the lesson of multiplication of polynomials a bit challenging and boring. Thanks to this article, because it will help you to understand how different types of polynomials are multiplied.  

Before jumping into multiplying polynomials, let’s recall what monomials, binomials and polynomials are.

A monomial is an expression with one term. Examples of monomial expression are 3x, 5y, 6z, 2x etc. Monomial expressions are multiplied the same way integers are multiplied.

A binomial is an algebraic expression with two terms which are separated by either addition sign (+) or subtraction sign (-). Examples of binomial expressions are 2x + 3, 3x – 1, 2x+5y, 6x−3y etc. Binomial expressions are multiplied using FOIL method. F-O-I- L is the short form of ‘first, outer, inner and last.’ The general formula of foil method is; (a + b) × (m + n) = am + an + bm + bn.

Let’s take a look at the example below.

Example 1

Multiply (x – 3) (2x – 9)

Solution

  • Multiply the first terms together;

= (x) * (2x) = 2x 2

  • Multiply the outermost terms of each binomial;

= (x) *(–9) = –9x

  • Multiply the inner terms of the binomials;

= (–3) * (2x) = –6x

  • Multiply the last terms of each binomial;

= (–3) * (–9) = 27

  • Sum up the products following the foil order and collect the like terms;

= 2x 2 – 9x -6x + 27

= 2x 2 – 15x +27

On the other hand, a polynomial is an algebraic expression that consist of one or more terms involving constants and variables with coefficients and exponents.

The terms in a polynomial are linked by addition, subtraction, or multiplication, but not division.

It is also important to note that, a polynomial can’t have fractional or negative exponents. Examples of polynomials are; 3y2 + 2x + 5, x3 + 2 x 2 − 9 x – 4, 10 x 3 + 5 x + y, 4x2 – 5x + 7) etc.

How to Multiply Polynomials?

To multiply polynomials, we use the distributive property whereby the first term in one polynomial is multiplied by each term in the other polynomial.

The resulting polynomial is then simplified by adding or subtracting identical terms. It should be noted that resulting polynomial has a higher degree than the original polynomials.

NOTE: To multiply variables, you multiply their coefficients and then add the exponents.

Multiplying a polynomial by a monomial

Let’s understand this concept with a help of a few examples below.

Example 2

Multiply x – y – z by -8x2.

Solution

Multiply each term of the polynomial x – y – z by the monomial -8x2.
⟹ -8x2 * (x – y – z)
= (-8x2 * x) – (-8x2 *y) – (-8x2 * z)

Add the like terms to get;
= -8x3 + 8x2y + 8x2z

Example 3

Multiply 4p3 – 12pq + 9q2 by -3pq.

Solution

= 3pq * (4p3 – 12pq + 9q2)

Multiply each term of the polynomial by the monomial
⟹ (-3pq * 4p3) – (-3pq * 12pq) + (-3pq * 9q2)
= 12p4q + 36p2q2 – 27pq3

Example 4

Find the product of 3x + 5y – 6z and – 5x

Solution

= -5x * (3x + 5y – 6z)

= (-5x * 3x) + (-5x * 5y) – (-5x * 6z)

= -15x2 – 25xy + 30xz

Example 5

Multiply x2 + 2xy + y2 + 1 by z.

Solution

= z * (x2 + 2xy + y2 + 1)

Multiply each term of the polynomial by the monomial
⟹ (z * x2) + (z * 2xy) + (z * y2) + (z * 1)
= x2z + 2xyz + y2z + z

Multiplying a polynomial by a binomial

Let’s understand this concept with a help of a few examples below.

Example 6

Multiply (a2 − 2a) * (a + 2b − 3c)

Solution

Apply the distributive law of multiplication

⟹ a2 * (a + 2b − 3c) − 2a * (a + 2b − 3c)

⟹ (a2 * a) + (a2 * 2b) + (a2 * −3c) − (2a * a) − (2a * 2b) − (2a * −3c)

= a3 + 2a2b − 3a2c − 2a2 − 4ab + 6ac

Example 7

Multiply (2x + 1) by (3x​2 − x + 4)

Solution

Use the distributive property to multiply the expressions;

⟹ 2x (3x2 − x + 4) + 1(3x2 – x + 4)
⟹ (6x3 − 2x2 + 8x) + (3x2 – x + 4)

Combine like terms.

⟹ 6x3 + (−2x2 + 3x2) + (8x − x) + 4

= 6x3 + x2 + 7x + 4

Example 8

Multiply (x + 2y) by (3x − 4y + 5)

Solution

= (x + 2y) * (3x − 4y + 5)

= 3x2 − 4xy + 5x + 6xy − 8y2 + 10y

= 3x2 + 2xy + 5x − 8y2 + 10y

Practice Questions

Find the product of the following pairs of expressions:

  1. 3ab3c and -2a3b2– 3a3c2 – 4b3c2
  2. axy and ax – yx + ay
  3. 5x and x + x2+ 1
  4. –6xy and 4x2– 5xy – 2y2
  5. 4x – 5 and 2x2 + 3x – 6
  6. 3x + 2 and 4x2– 7x + 5
  7. 3x2 and 4x2– 5x + 7
  8. 3x2– 2x2y + 9y2 and –y2
  9. 10ab and ab + bc + ca
  10. -11ab2c and 5ab + 2bc – 4ca

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