Answer:
x² -3x +2
Explanation:
You want the expanded form of the product (x -2)(x -1).
Simplify
As you know, "FOIL" is an acronym to help you form the partial product terms of the product of two binomials. It is fully equivalent to applying the distributive property 3 times. Whether you call it the distributive property, or FOIL is up to you:
(x -2)(x -1)
= x(x -1) -2(x -1) . . . . . . . apply the distributive property once
= x² -x -2(x -1) . . . . . . . . apply the distributive property again
= x² -x -2x +2 . . . . . . . . apply the distributive property a third time
These terms are the products of the first terms (x)(x), the outer terms (x)(-1), the inner terms (-2)(x), and the last terms (-2)(-1). These are the same partial products you get using FOIL.
= x² -3x +2 . . . . . combining like terms
Alternate approach
You recognize that the leading terms are both degree 1, so the their product will have degree 1+1 = 2. In general the product of polynomials of degree 1 will have terms of all degrees up to 2.
The degree 2 term of the result can only be formed as the product of the degree 1 terms in the factors:
(x)(x) = x² . . . . the degree 2 term of the result
The degree 1 term of the result will be formed by multiplying the degree 1 term in any factor by the product of the degree 0 terms in any remaining factors. It will be the sum of such products:
(x)(-1) +(-2)(x) = -3x . . . . the degree 1 term of the result
The degree 0 term of the result will be the product of the constants in any of the factors:
(-2)(-1) = +2 . . . . the degree 0 term of the result
The whole result is the sum of these terms:
= x² -3x +2
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Additional comment
This same approach of identifying the contributors to any product term can be used for multiplying polynomials of any degree. When the factors are polynomials of the same length, there is a nice pattern to the terms that are multiplied to give the terms of the result.
For example, the product of two quadratics with coefficients ...
will have 5 terms with coefficients ...
(ad), (ae+db), (ag+dc+be), (bg+ec), (cg)
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