How would I factor this?
(x-3)^3+ (3x-2)^3
Can someone explain this to me step by step? thanks
(1) Multiply them out.
(2) Combine terms with the same exponent of x
(3) Try to factor what is left
The last step will probably be the hardest, but you need to get there first.
dont you have a formula for the factors of two cubes?
a^3+b^3= (a+b)(a^2-ab+b^2)
Here, a is x-3 and b is 3x-2.
To factor the given expression, we can use a special factoring formula called the sum of cubes. The formula states that:
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
In this case, we have:
(x-3)^3 + (3x-2)^3
Let's use the sum of cubes formula by identifying a and b in each term:
a = (x - 3)
b = (3x - 2)
Now, let's plug in the values into the formula:
(x-3)^3 + (3x-2)^3 = (a + b)(a^2 - ab + b^2)
= [(x - 3) + (3x - 2)][(x - 3)^2 - (x - 3)(3x - 2) + (3x - 2)^2]
Simplifying further:
[(x - 3) + (3x - 2)][x^2 - 6x + 9 - (3x^2 - 11x + 6) + 9x^2 - 12x + 4]
Combine like terms within each bracket:
(4x - 5)(-3x^2 - 11x + 19)
And there you have it! The factored form of the expression is (4x - 5)(-3x^2 - 11x + 19).