Polynomials are expressions containing finite number of terms. None of the terms can have a division by a variable and also the exponents of any term must only be a non-negative integer. These types of expressions can be added, subtracted, multiplied or can be divided.
In this lesson let discuss about multiplying polynomials. The method of how to multiply polynomials is based on the concept of distributive property. Please express your views of this topic Operations with Polynomials by commenting on blog.
A polynomial with only terms is called as binomials. We all know that two binomials are multiplied by the technique FOIL. The same concept is slightly modified and extended in case of polynomial multiplications. As one of the multiplying polynomials examples, let us consider the following with just one variable. (a0xn + a1xn-1 + a2xn-2 + ….. + an-1x + an)* (b0xm + b1xm-1 + b2xm-2 + ….. + bm-1x + bm) = ?
Pick up the first term a0xn of the first expression and multiply that with all the terms of the second expression and add the products as per the distributive property of multiplication over addition. This is first set of expression of the entire product. Now take the second term a1xn-1 of the first expression and repeat the same process.
The result will be the second set of the expression for the entire product. The method is repeated till the last term of the first expression is multiplied with all the terms of the second expression. This is the final set of expression of the entire product. Now add all sets of expressions and simplify the sum by algebraically adding like terms. You may notice that the degree of the final product is sum of the degrees of the given expressions. Is this topic What is an Algebraic Expression? hard for you? Watch out for my coming posts.
For better clarity let us take an actual case in multiplying polynomials problems.
(3x^2 – 2x + 4)*( x^3 + 2x^2 – x + 5) = ?
Step 1: (3x^2)*( x^3 + 2x^2 – x + 5) = 3x5 + 6x^4 – 3x^3 + 15x^2
Step 2: (-2x)*( x^3 + 2x^2 – x + 5) = -2x^4 – 4x^3 + 2x^2 – 10x
Step 3: (4)*( x^3 + 2x^2 – x + 5) = 4x^3 + 8x^2 – 4x + 20
Adding the expressions obtained in all the steps, we can say that
(3x^2 – 2x + 4)*( x^3 + 2x^2 – x + 5) = 3x5 + 6x^4 – 3x^3 + 15x^2- 2x^4 – 4x^3 + 2x^2 – 10x + 4x^3 + 8x^2– 4x + 20
Simplifying by adding the like terms, the final product can be written as,
3x5 + 4x^4 – 3x^3 + 25x^2 - 14x + 20
The given expressions had the degrees as 2 and 3 respectively and it may be seen the degree of the the product is 5 which is 2 + 3.
In this lesson let discuss about multiplying polynomials. The method of how to multiply polynomials is based on the concept of distributive property. Please express your views of this topic Operations with Polynomials by commenting on blog.
A polynomial with only terms is called as binomials. We all know that two binomials are multiplied by the technique FOIL. The same concept is slightly modified and extended in case of polynomial multiplications. As one of the multiplying polynomials examples, let us consider the following with just one variable. (a0xn + a1xn-1 + a2xn-2 + ….. + an-1x + an)* (b0xm + b1xm-1 + b2xm-2 + ….. + bm-1x + bm) = ?
Pick up the first term a0xn of the first expression and multiply that with all the terms of the second expression and add the products as per the distributive property of multiplication over addition. This is first set of expression of the entire product. Now take the second term a1xn-1 of the first expression and repeat the same process.
The result will be the second set of the expression for the entire product. The method is repeated till the last term of the first expression is multiplied with all the terms of the second expression. This is the final set of expression of the entire product. Now add all sets of expressions and simplify the sum by algebraically adding like terms. You may notice that the degree of the final product is sum of the degrees of the given expressions. Is this topic What is an Algebraic Expression? hard for you? Watch out for my coming posts.
For better clarity let us take an actual case in multiplying polynomials problems.
(3x^2 – 2x + 4)*( x^3 + 2x^2 – x + 5) = ?
Step 1: (3x^2)*( x^3 + 2x^2 – x + 5) = 3x5 + 6x^4 – 3x^3 + 15x^2
Step 2: (-2x)*( x^3 + 2x^2 – x + 5) = -2x^4 – 4x^3 + 2x^2 – 10x
Step 3: (4)*( x^3 + 2x^2 – x + 5) = 4x^3 + 8x^2 – 4x + 20
Adding the expressions obtained in all the steps, we can say that
(3x^2 – 2x + 4)*( x^3 + 2x^2 – x + 5) = 3x5 + 6x^4 – 3x^3 + 15x^2- 2x^4 – 4x^3 + 2x^2 – 10x + 4x^3 + 8x^2– 4x + 20
Simplifying by adding the like terms, the final product can be written as,
3x5 + 4x^4 – 3x^3 + 25x^2 - 14x + 20
The given expressions had the degrees as 2 and 3 respectively and it may be seen the degree of the the product is 5 which is 2 + 3.
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