Write each expression as the product of two binomials.
1) a^3 - 3a^2 + 3a - 9
2) 2x^2 - 3x^2 - 4x + 6
use grouping
the first:
a^3 - 3a^2 + 3a - 9
= a^2(a - 3) + 3(a - 3)
= (a - 3)(a^2 + 3)
do the 2nd the same way.
To write each expression as the product of two binomials, we need to factor the given expressions.
1) a^3 - 3a^2 + 3a - 9:
To factor this expression, we look for common factors. In this case, we find that 1 is a common factor. Thus, we can write the expression as:
a^3 - 3a^2 + 3a - 9 = (a^3 - 3a^2) + (3a - 9)
Next, we factor out the common terms from each group:
a^3 - 3a^2 = a^2(a - 3)
3a - 9 = 3(a - 3)
Putting it all together, we have:
a^3 - 3a^2 + 3a - 9 = (a^2 + 3)(a - 3)
2) 2x^2 - 3x^2 - 4x + 6:
To factor this expression, we group the terms with like variables together:
(2x^2 - 3x^2) + (-4x + 6)
Now we factor out the common factors from each group:
2x^2 - 3x^2 = x^2(2 - 3) = -x^2
-4x + 6 = -2(2x - 3)
Putting it all together, we have:
2x^2 - 3x^2 - 4x + 6 = -x^2 - 2(2x - 3)
So, the expressions as the product of two binomials are:
1) a^3 - 3a^2 + 3a - 9 = (a^2 + 3)(a - 3)
2) 2x^2 - 3x^2 - 4x + 6 = -x^2 - 2(2x - 3)