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# Multiplying Polynomials â€“ Explanation & Examples

Many students will find the lesson of **multiplication of polynomials** a bit challenging and boring. This article will help you to understand how different types of polynomials are multiplied.

Before jumping into multiplying polynomials, letâ€™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 separated by either the addition sign (+) or subtraction sign (-). Examples of binomial expressions are 2*x*Â + 3, 3*x* â€“ 1, 2x+5y, 6xâˆ’3y, etc. Binomial expressions are multiplied using the FOIL method. F-O-I- L is the short form of â€˜first, outer, inner and last.â€™ The general formula of foil method is; (a + b) Ã— (m + n) = am + an + bm + bn.

Letâ€™s take a look at the example below.

*Example 1*

Multiply (xÂ â€“ 3) (2xÂ â€“ 9)

__Solution__

- Multiply the first terms together;

= (x) * (2x) = 2x ^{2}

- Multiply the outermost terms of each binomial;

= (*x*) *(â€“9) = â€“9*x*

- Multiply the inner terms of the binomials;

= (â€“3) * (2*x*) = â€“6*x*

- Multiply the last terms of each binomial;

= (â€“3) * (â€“9) = 27

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

= 2x ^{2} â€“ 9x -6x + 27

= 2x ^{2} â€“ 15x +27

**On the other hand, a polynomial** is an algebraic expression consisting 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 canâ€™t have fractional or negative exponents. Examples of polynomials are; 3y^{2}Â + 2x + 5, x^{3}Â + 2Â xÂ ^{2}Â âˆ’ 9Â xÂ â€“ 4, 10Â xÂ ^{3}Â + 5Â xÂ +Â y, 4x^{2}Â â€“ 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. You should note that the 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

Letâ€™s understand this concept with a help of a few examples below.

*Example 2*

Multiply x â€“ y – z by -8x^{2}.

__Solution__

Multiply each term of the polynomial x â€“ y – z by the monomial -8x^{2}.

âŸ¹ -8x^{2}Â * (x â€“ y â€“ z)

= (-8x^{2}Â * x) â€“ (-8x^{2}Â *y) â€“ (-8x^{2}Â * z)

Add the like terms to get;

= -8x^{3}Â + 8x^{2}y + 8x^{2}z

*Example 3*

Multiply 4p^{3}Â â€“ 12pq + 9q^{2}Â by -3pq.

__Solution__

= 3pq * (4p^{3}Â â€“ 12pq + 9q^{2})

Multiply each term of the polynomial by the monomial

âŸ¹ (-3pq * 4p^{3}) – (-3pq * 12pq) + (-3pq * 9q^{2})

= 12p^{4}q + 36p^{2}q^{2}Â â€“ 27pq^{3}

*Example 4*

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

__Solution__

= -5x * (3x + 5y â€“ 6z)

= (-5x * 3x) + (-5x * 5y) â€“ (-5x * 6z)

= -15x^{2}Â â€“ 25xy + 30xz

*Example 5*

Multiply x^{2}Â + 2xy + y^{2}Â + 1 by z.

__Solution__

= z * (x^{2}Â + 2xy + y^{2}Â + 1)

Multiply each term of the polynomial by the monomial

âŸ¹ (z * x^{2}) + (z * 2xy) + (z * y^{2}) + (z * 1)

= x^{2}z + 2xyz + y^{2}z + z

### Multiplying a polynomial by a binomial

Letâ€™s understand this concept with a help of a few examples below.

*Example 6*

Multiply (a^{2}Â âˆ’ 2a) * (a + 2b âˆ’ 3c)

__Solution__

Apply the distributive law of multiplication

âŸ¹ a^{2}Â * (a + 2b âˆ’ 3c) âˆ’ 2a * (a + 2b âˆ’ 3c)

âŸ¹ (a^{2}Â * a) + (a^{2}Â * 2b) + (a^{2}Â * âˆ’3c) âˆ’ (2a * a) âˆ’ (2a * 2b) âˆ’ (2a * âˆ’3c)

= a^{3}Â + 2a^{2}b âˆ’ 3a^{2}c âˆ’ 2a^{2}Â âˆ’ 4ab + 6ac

*Example 7*

Multiply (2x + 1) by (3xâ€‹^{2 }âˆ’ x + 4)

__Solution__

Use the distributive property to multiply the expressions;

âŸ¹ 2x (3x^{2 }âˆ’ x + 4) + 1(3x^{2 }â€“ x + 4)

âŸ¹ (6x^{3 }âˆ’ 2x^{2 }+ 8x) + (3x^{2 }â€“ x + 4)

Combine like terms.

âŸ¹ 6x^{3 }+ (âˆ’2x^{2 }+ 3x^{2}) + (8x âˆ’ x) + 4

= 6x^{3 }+ x^{2 }+ 7x + 4

*Example 8*

Multiply (x + 2y) by (3x âˆ’ 4y + 5)

__Solution__

= (x + 2y) * (3x âˆ’ 4y + 5)

= 3x^{2}Â âˆ’ 4xyÂ + 5xÂ + 6xyÂ âˆ’ 8y^{2}Â + 10y

= 3x^{2}Â + 2xyÂ + 5x âˆ’ 8y^{2}Â + 10y