In this lesson on Multiplying Polynomials, we will learn:
Pre-requisites for understanding this multiplying polynomials lesson:
The following rules for multiplying positive and negative numbers:
Multiplication Rule 1:
– × + = –
Example 1:
– a × +b = – ab
Multiplication Rule 2:
× – = +
Example 2:
Multiplication Rule 3:
+ × – = – ab
Example 3:
+ a × – b= – ab
Multiplication Rule 4:
+ × + = +
Example 4:
+a × + b = + ab
Multiplying polynomials is one of the four fundamental operations of algebraic expressions.
The other three fundamental operations on algebraic expressions are:
Now, let us learn Multiplying Polynomials by remembering the following multiplication rules (for multiplying polynomials)
Step 1: Multiply the Numerical Coefficients separately
Step 2: Multiply the literal coefficients separately and
Step 3: Multiply the above two products (the numerical coefficients product and the literal coefficients product)
SOLVED PROBLEM NO.1:
Multiply the following two monomials:
3a^{2} and 5a^{3}
Step 1: Numerical coefficients product = 3 × 5 = 15
Step 2: literal coefficients product = a^{2} × a^{3 }= a^{5} (from the law of exponents – if bases are same, then add the powers)
Step 3:
Now, write the product of the numerical coefficients (15) and literal coefficients (a5), i.e.
15 × a^{5} = 15a^{5}.
Note:
15a^{5} stands for the product of 15 and a^{5}, just as
pq means product of p and q i.e. p × q, and
3a means 3 × a.
Now, Multiplying a Monomial with a Binomial
SOLVED PROBLEM NO 2:
Multiply the Monomial 3a^{2} with the binomial x + y
Solution:
First set up the product as below:
3a^{2} × (x + y)
Now, Recall the distributive property a (b + c) which is:
a(b + c) = ab + ac
So, by applying the above distributive property to 3a^{2} × (x + y), we get
3a^{2}x + 3a^{2}y.
SOLVED PROBLEM NO 3:
Perform the following multiplication of polynomials:
3a^{2} (5a^{3}b + 6ac)
Solution:
From the distributive property a (b + c) = ab + ac,
We get the following two terms:
3a^{2} × 5a^{3}b + 3a^{2} × 6ac
Now from exponents rules, add powers of same bases and applying the following steps:
Step 1: Multiply the Numerical Coefficients separately
Step 2: Multiply the literal coefficients separately and
We get:
3a^{2} × 5a^{3}b = 3×5×a^{2} ×a^{3} × b = 15 a^{5}×b and
3a^{2} × 6ac = 3 × 6 × a^{2} × a × c = 18a^{3}c, so finally the product of
3a^{2} and (5a^{3}b + 6ac) is 15 a^{5}×b + 18a^{3}c.
MATH SKILLS RELATED TO MULTIPLYING POLYNOMIALS
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