assignment 5.multiplying polynomials

Multiplying Polynomials

A polynomial looks like this:

To multiply two polynomials:

• multiply each term in one polynomial by each term in the other polynomial
• add those answers together, and simplify if needed

Let us look at the simplest cases first.

1 term × 1 term   (monomial times monomial)

To multiply one term by another term, first multiply the constants , then multiply each variable together and combine the result, like this (press play):

(Note: I used "·" to mean multiply. In Algebra we don't like to use "×" because it looks too much like the letter "x")

For more about multiplying terms, read Multiply and Divide Variables with Exponents

1 term × 2 terms   (monomial times binomial)

Multiply the single term by each of the two terms, like this:

2 term × 1 terms   (binomial times monomial)

Multiply each of the two terms by the single term, like this:

(I did that one a bit faster by multiplying in my head before writing it down)

2 terms × 2 terms (binomial times binomial)

That is 4 different multiplications ... Why?

It is the same when we multiply binomials!

Instead of Alice and Betty, let's just use a and b , and Charles and David can be c and d :

We can multiply them in any order so long as each of the first two terms gets multiplied by each of the second two terms .

But there is a handy way to help us remember to multiply each term called " FOIL ".

It stands for " F irsts, O uters, I nners, L asts":

So you multiply the "Firsts" (the first terms of both polynomials), then the "Outers", etc.

Let us try this on a more complicated example:

2 terms × 3 terms (binomial times trinomial)

"FOIL" won't work here, because there are more terms now. But just remember:

Multiply each term in the first polynomial by each term in the second polynomial

And always remember to add Like Terms :

Example: (x + 2y)(3x − 4y + 5)

(x + 2y)(3x − 4y + 5)

= 3x 2 − 4xy + 5x + 6xy − 8y 2 + 10y

= 3x 2 + 2xy + 5x − 8y 2 + 10y

Note: −4xy and 6xy are added because they are Like Terms.

Also note: 6yx means the same thing as 6xy

Long Multiplication

You may also like to read about Polynomial Long Multiplication

Multiplying Polynomials

In these lessons, we will learn how to multiply polynomials.

Related Pages More Lessons on Algebra Free Math Worksheets

Multiplying Polynomials and Monomials

When finding the product of a monomial and a polynomial, we multiply the monomial by each term of the polynomial. Be careful with the sign (+ or –) of each term.

Example: Evaluate a) 5( x + y ) b) – 2 x ( y + 3) c) 5 x ( x 2 – 3) d) –2 x 3 ( x 2 – 3 x + 4)

Solution: a) 5( x + y ) = 5 x + 5 y b) – 2 x ( y + 3) = – 2 xy – 6 x c) 5 x ( x 2 – 3) = 5 x 3 – 15 x d) –2 x 3 ( x 2 – 3 x + 4) = –2 x 5 + 6 x 4 – 8 x 3

Multiplying Polynomials and Polynomials

The multiplication of polynomials having more than one term requires the repeated use of the distributive property.

Example: Multiply (2 x + 5)( x +1)

Solution: (2 x + 5)( x + 1) = x (2 x + 5) + 1(2 x + 5) = 2 x 2 + 5 x + 2 x + 5 = 2 x 2 + 7 x + 5

Example: Multiply (5 y 2 – 2 y + 3)(3 y – 4)

Solution: (5 y 2 – 2 y + 3)(3 y – 4) = 3 y (5 y 2 – 2 y + 3) – 4(5 y 2 – 2 y + 3) = 15 y 3 – 6 y 2 + 9 y – 20 y 2 + 8 y – 12 = 15 y 3 – 26 y 2 + 17 y – 12

Multiplying Polynomials – Special Products

We will now consider two special types of products of binomials. The first type is a trinomial that comes from squaring a binomial. (It is also called a perfect square) Its general form is ( A + B ) 2 = A 2 + 2 AB + B 2

Example: ( x + 5) 2 = x 2 + 10 x + 25 ( y – 3) 2 = y 2 – 6 y + 9 The second type is a binomial that comes from multiplying two binomials. (It is also called the difference of two squares) Its general form is ( A + B )( A – B ) = A 2 – B 2

Example: ( x + 4)( x – 4) = x 2 – 16 (5 y – 7)(5 y + 7) = 25 y 2 - 49

Multiplying Monomials and/or Binomials and FOIL We multiply monomials and binomials using different methods, including the distributive property and FOIL. FOIL is a mnemonic device to remember how to find the product of two binomials: we multiply the First, Outer, Inner, and then Last terms in each binomial. When multiplying monomials and binomials, it is important to remember the rules of multiplying exponents.

How to multiply a binomial and a trinomial by distributing each of the terms in the binomial through the trinomial?

Multiplying polynomials Special products: Difference of Two Squares and Perfect Square.

Multiplying Polynomials This video explains how to multiply monomials and polynomials.

Multiplying Polynomials - Slightly Harder Examples #1

Multiplying Polynomials - Slightly Harder Examples #2 Multiply a trinomial by a trinomial.

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Multiplying Polynomials by Polynomials

Examples, practice problems and steps!

Worksheet on this topic Multiply Polynomial Calc

Let's multiply the polynomial $$(3x^6 + 2x^5 + 5)$$ by the polynomial (5x + 2) .

Distribute .

Add the resulting Polynomials

$$( 15x^7 + 10x^6 + 25x) + (6x^6 + 4x^5 + 10) = 15x^7 + 16x^6 + 4x^6 + 25x + 10 $$

$$( 15x^7 + 10x^6 + 25x) + (6x^6 + 4x^5 + 10) \\ = 15x^7 + 16x^6 + 4x^6 + 25x + 10 $$

Let's multiply the polynomial $$(3x^6 + 2x^5 + 5)$$ by $$(4x^2 + x + 5)$$ .

Add the resulting Polynomials .

$$ (12x^8 + 8x^7 + 20x^2) + (3x^7 + 2x^6 + 5x) + (15x7^6+ 10x^5 + 25) \\ = \boxed{ 12x^8 + 11x^7 + 17x^6 + 10x^5 + 20x^2 + 5x + 25} $$

Practice Problems

Multiply the polynomial (3x + 4) by the polynomial (5x 4 + 7x 3 + 5x) .

This is similar to example 1 .

15x 5 + 41x 4 + 28x 3 + 15x 2 + 20x

Multiply (2x 7 + 4x 2 + 3x)(3x 8 + 2x 3 + 15x) .

This is similar to example 2 .

6x 13 + 4x 12 + 12x 8 + 27x 7 + 6x 6 + 20x 2 + 15x

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• Math Article

Multiplying Polynomials

Multiplying polynomials is a basic concept in algebra. Multiplication of two polynomials will include the product of coefficients to coefficients and variables to variables. We can easily multiply polynomials using rules and following some simple steps. Let us learn more about multiplying polynomials with examples in this article.

Rules for Multiplying Polynomials

Multiplying polynomials require only three steps.

• First, multiply each term in one polynomial by each term in the other polynomial using the distributive law.
• Add the powers of the same variables using the exponent rule.
• Then, simplify the resulting polynomial by adding or subtracting the like terms.

It should be noted that the resulting degree after multiplying two polynomials will be always more than the degree of the individual polynomials.

Multiplying Polynomials Using Exponent Law

If the variable is the same but has different exponents of the given polynomials, then we need to use the exponent law.

Example: Multiply 2x 2  × 3x

Here, the coefficients and variables are multiplied separately.

= (2 × 3) × (x 2  × x)

= 6 × x 2+1

Multiplying Polynomials having different variables

Follow the below-given steps for multiplying polynomials:

• Step 1: Place the two polynomials in a line.

For example, for two polynomials, (6x−3y) and (2x+5y), write as: (6x−3y)×(2x+5y)

• Step 2: Use distributive law and separate the first polynomial.
• Step 3: Multiply the monomials from the first polynomial with each term of the second polynomial.

⇒ [6x × (2x+5y)] − [3y × (2x+5y)] = (12x 2 +30xy) − (6yx+15y 2 )

• Step 4: Simplify the resultant polynomial, if possible.

⇒ (12x 2 +30xy) − (6yx+15y 2 ) = 12x 2 +24xy−15y 2

Degree – Multiplying Polynomials

For two polynomials equations, P and Q, the degree after multiplication will always be higher than the degree of P or Q. The degree of the resulting polynomial will be the summation of the degree of P and Q.

Multiplying Polynomials by Polynomials

It is known that there are different types of polynomial based on their degree like monomial, binomial, trinomial, etc. The steps to multiply polynomials are the same for all types.

Multiplying Monomial by Monomial

A monomial is a single term polynomial. If two or more monomials are multiplied together, then the resulting product will be a monomial.

Examples are:

• 5x × 7x = 5 × x × 7 × x = 35x 2
• 2x × 3y × 4z = 2 × x × 3 × y × 4 × z = (2 × 3 × 4) × (x × y × z) = 24xyz

Multiplying Binomial by a Binomial

A binomial is a two-term polynomial. When a binomial is multiplied by a binomial, the distributive law of multiplication is followed.

We know that Binomial has 2 terms. Multiplying two binomials give the result having a maximum of 4 terms (only in case when we don’t have like terms). In the case of like terms, the total number of terms is reduced.

Like Terms:

According to the commutative law of multiplication, terms like ‘ab’ and ‘ba’ give the same result. Thus they can be written in both forms.

For example, 5×6 = 6×5 = 30

Now, Consider two binomials given as (a+b) and (m+n).

Multiplying them we have,

(a+b)×(m+n)

⇒ a×(m+n)+b×(m+n) (Distributive law of multiplication)

⇒ (am+an)+(bm+bn) (Distributive law of multiplication)

Solved Examples

Example 1: Find the result of multiplication of two polynomials (6x +3y) and (2x+ 5y).

Solution- (6x−3y)×(2x+5y)

⇒6x×(2x+5y)−3y×(2x+5y) (Distributive law of multiplication)

⇒(12x 2 +30xy)−(6yx+15y 2 ) (Distributive law of multiplication)

⇒12x 2 +30xy−6xy−15y 2 (as xy = yx)

Thus, (6x+3y)×(2x+5y)=12x 2 +24xy−15y 2

Let us take up an example. Say, you are required to multiply a binomial (5y + 3z) with another binomial (7y − 15z). Let us see how it is done.

(5y + 3z) × (7y − 15z) = 5y × (7y − 15z) + 3z × (7y − 15z) (Distributive law of multiplication)

= (5y × 7y) − (5y × 15z) + (3z × 7y) − (3z × 15z) (Distributive law of multiplication)

= 35y 2 − 75yz + 21zy − 45z 2

= 35y 2 − 75yz + 21yz − 45z 2

As, (yz = zy)

(5y + 3z) × (7y − 15z) = 35y 2 −54yz − 45z 2

Multiplying Binomial with a Trinomial

A trinomial is a three-term polynomial. When multiplying polynomials, that is, a binomial by a trinomial, we follow the distributive law of multiplication. Thus, 2 × 3 = 6 terms are expected to be in the product. Let us take up an example.

(a 2 − 2a) × (a + 2b − 3c)

= a 2 × (a + 2b − 3c) − 2a × (a + 2b − 3c) (Distributive law of multiplication)

= (a 2 × a) + (a 2 × 2b) + (a 2 × −3c) − (2a × a) − (2a × 2b) − (2a × −3c) (Distributive law of multiplication)

= a 3 + 2a 2 b − 3a 2 c − 2a 2 − 4ab + 6ac

Now, by rearranging the terms,

(a 2 − 2a) × (a + 2b − 3c) = a 3 − 2a 2 + 2a 2 b − 3a 2 c− 4ab + 6ac

Important Facts

When multiplying polynomials, the following pointers should be kept in mind:

• Distributive Law of multiplication is used twice when 2 polynomials are multiplied.
• Look for the like terms and combine them. This may reduce the expected number of terms in the product.
• Preferably, write the terms in the decreasing order of their exponent.
• Be very careful with the signs when you open the brackets.

Related Articles

Practice Questions

• Multiply 2x by 3y.
• Multiply (3x – a) (4x – y)
• Find the product of (x + 2y)(3x − 4y + 5)
• Multiply (x – 3) (2x – 9)
• What is the product of 3x 2 and 4x 2 – 5x + 7?

Frequently Asked Questions – FAQs

How to multiply polynomials, how to multiply binomials, how to multiply large polynomials.

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