factor the expression a^2-6ab+9b^2 into a product of binomials
To factor the expression \(a^2 - 6ab + 9b^2\) into a product of binomials, we can use the FOIL method.
The FOIL method involves multiplying the first terms, the outer terms, the inner terms, and the last terms of two binomials. In this case, the binomials will have the form \(a + nb\) and \(a + mb\) where \(n\) and \(m\) are integers.
So, let's proceed with the factoring:
Step 1: Multiply the first terms: multiply \(a\) and \(a\). The result is \(a \cdot a = a^2\).
Step 2: Multiply the last terms: multiply \(nb\) and \(mb\). The result is \(nb \cdot mb = nmb^2\).
Step 3: Find the factors of the middle term (-6ab) that when multiplied, give the result from Step 2 (nmb^2). In this case, the only combination that works is -3 and -3: (-3ab) and (-3ab) which give (-3ab) * (-3ab) = 9a^2b^2.
Step 4: Rewrite the middle term (-6ab) using the factors from Step 3: -3ab - 3ab = -6ab.
Step 5: Assemble the binomial factors: \(a^2 - 6ab + 9b^2\) can be factored as \((a - 3b)(a - 3b)\).
Alternatively, we can write it in simplified form: \((a - 3b)^2\).
So, the expression \(a^2 - 6ab + 9b^2\) factors into a product of binomials as \((a - 3b)(a - 3b)\) or \((a - 3b)^2\).