How do you factor this: f(x)=1/2(x-2)^3+1

If you mean 1/2 * (x-2)^3 + 1 then we have that pesky 1/2 to deal with.

(a^3 + b^3) = (a+b)(a^2 - ab + b^2)

So, let a = cbrt(1/2)*x and b = 1, or 2^(1/3)*x

Then 1/2 * (x-2)^3 + 1
= (x*2^1/3 + 1)(x^2 * 2^2/3 - x*2^1/3 + 1)
or,
(cbrt(2)x + 1)(cbrt(4)x^2 - cbrt(2)x + 1)

Oops. Those 2's in the solution should all be 1/2.

Can you explain it more clearly?

What does cbrt mean?

sqrt = square root

cbrt = cube root.

The fractional exponents get clumsy

suppose you had

125x^3 + 1
that is (5x)^3 + 1^3
so it factors into
(5x+1)((5x)^2 - (5x)*1 + 1^2)
= (5x+1)(25x^2 - 5x + 1)

You have (1/2)x^3 which is ((1/2)^(1/3)x)^3 or (cbrt(1/2)x)^3

To factor the given equation, f(x) = 1/2(x-2)^3 + 1, we need to simplify and express it in a factored form.

Step 1: Expand the expression
Start by expanding the cube term. Using the binomial expansion formula for (a - b)^3, we have:
(x - 2)^3 = (x - 2)(x - 2)(x - 2)
Expanding this expression, we get:
(x - 2)^3 = (x - 2)(x^2 - 4x + 4)
Now substitute this back into the original equation:
f(x) = 1/2(x^2 - 4x + 4) + 1

Step 2: Distribute
Distribute the 1/2 to each term inside the brackets:
f(x) = 1/2x^2 - 2x + 2 + 1

Step 3: Combine like terms
Combine the constant terms:
f(x) = 1/2x^2 - 2x + 3

The factored form of the equation f(x) = 1/2(x-2)^3 + 1 is f(x) = 1/2x^2 - 2x + 3.