subject: Expanding Polynomials [print this page] Expanding polynomial is nothing but multiplying factors. The expansion of polynomial is done with non-expanded polynomial. There are two types of non-expanded polynomial. In which one is having factors like (ax + b) (cx + d), where x is a variable and a, b, c and d are constants and the other is the factor will have some powers like (ax + b) 2. Most commonly the method used for expanding polynomials is FOIL method.
Steps for Expanding Polynomials:
There are eight steps for expanding polynomials. They are following,
Step 1: First write the factors inside the parenthesis.
Step 2: If one of the polynomials is shorter than the other then write the small polynomial as a first one.
Step 3: Now take the first term from first polynomial and multiply it with the each term of the second polynomial.
Step 4: Now take the next term from the first polynomial and multiply it with the each term of the second polynomial.
Step 5: Repeat the step 4 until the end of the last term.
Step 6: Now add all the solution which we got in previous steps.
Step 7: Combine like terms in expression.
Step 8: Simplify the result by combining common terms.
Example Problems on Expanding Polynomials:
Problem 1:
Expanding polynomials: (x+3) (x+5)
Solution:
Step 1: Given, factors are (x+3) (x+5)
Step 2: Here order is correct form. Small polynomial is first one.
This can expand by using FOIL Method.
Step 3: Multiply first term of first polynomial to each term of second polynomial.
Firsts '->' x * x '->' x ^ 2
Inner '->' x * 5 '->' 5x
Step 4: Multiply next term of first polynomial to each term of second polynomial.
Outer '->' 3 * x '->' 3x
Lasts '->' 3 * 5 '->' 15
Step 5: Now sum all solutions '->' x^2+5x+3x+15
Step 6: Now combine the like terms '->' x^2+8x+15. (5x + 3x = 8x)
Step 7: Expanding polynomials for (x+3) (x+5) is x^2+8x+15.
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Problem 2:
Expanding polynomials: (x^3+2x^2+4x+8) (x+2)
Solution:
Given, the factors are (x^3+2x^2+4x+8) (x+2)
Step 1: Rewrite the given as (x+2) (x^3+2x^2+4x+8)
Step 2: Multiply first term of first polynomial to each term of second polynomial.
X * (x^3+2x^2+4x+8) x^4+2x^3+4x^2+8x
Step 3: Multiply second term of first polynomial to each term of second polynomial.
2 * (x^3+2x^2+4x+8) 2x^3+4x^2+8x+16
Step 4: Add the solutions of step 3 and 4.
(X^4+2x^3+4x^2+8x) + (2x^3+4x^2+8x+16)
Step 5: Combine the like terms.
X^4+2x^3+2x^3+4x^2+4x^2+8x+8x+16
Step 6: Add the like terms and simplify it.
X^4+4x^3+8x^2+16x+16.
Step 7: Expanding polynomials for (x+2) (x^3+2x^2+4x+8) is x^4+4x^3+8x^2+16x+16.
