The distributive property states that:
a(b + c) = ab + ac
In other words, when a number is multiplied by a set of parentheses, it is multiplied by each number within that set of parentheses. With numbers this is easy to see:
2(1 + 3) = 2 * 1 + 2 * 3 = 2 + 6 = 8
The distributive property becomes more useful when variables are involved:
3x(x + 4) = 3x*x + 3x*4 = 3x^2 + 12x
The distributive property allows you to simplify an expression with multiple parts by removing the factor all the parts have in common and putting what is left in parentheses:
3x + 12 = 3(x + 4)
In the above example, each part of the expression was divided by 3 and left the results in parentheses multiplied by 3. It is possible to simplify a longer expression in the same way. Simply remove the factor common to all the parts and put what is left in parentheses:
2x^2 + 12x + 6 = 2(x^2 + 6x + 3)
To multiply a polynomial with a single number or term, you can use the distributive property to translate the one big multiplication problem into a few little ones. Simply multiply the term outside the parentheses by every term within the parentheses:
3(x^2 + 4x - 7) = 3*x^2 + 3*4x - 3*7 = 3x^2 + 12x + 21
This is the reverse of the process used above to simplify a polynomial.
Multiplying two polynomials takes a bit more concentration, because instead of one number outside the parentheses, it's another set of parentheses. Start by breaking up the first set of parentheses, multiplying the second set by each term in the first:
(x + 3)(x + 5) =
x(x + 5) + 3(x + 5)
Next, multiply out each of the remaining sets of parentheses as above, and complete the remaining little multiplication problems:
x*x + x*5 + 3*x + 3*5 =
x^2 + 5x + 3x + 15 =
x^2 + 8x + 15