Expressing the polynomial 64x^3 - y^3 - 8z^3 + 24xyz in factor form

To express the given polynomial in factor form, we can use the factor theorem and factor out common factors. Let's break down the process step by step.

Step 1: Identify common factors

We observe that the given polynomial has common factors in each term. The common factor in this case is 8. By factoring out the common factor, we can simplify the polynomial.

Step 2: Factor out the common factor

Dividing each term of the polynomial by 8, we get:

64x^3 - y^3 - 8z^3 + 24xyz = 8(8x^3 - y^3 - z^3 + 3xyz)

Now we have a simplified polynomial that can be further factored.

Step 3: Factor the remaining polynomial

The polynomial 8x^3 - y^3 - z^3 + 3xyz can be further factored using the formula for the difference of cubes:

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

Applying this formula, we can factor the remaining polynomial as follows:

8x^3 - y^3 - z^3 + 3xyz = (2x - y)(4x^2 + 2xy + y^2) - z^3 + 3xyz

Now we have factored the polynomial into two terms.

Step 4: Simplify the remaining terms

The remaining terms - z^3 + 3xyz cannot be factored further, but we can rearrange them to make it more visually appealing:

-z^3 + 3xyz = -z^3 + 3xyz

Step 5: Final factor form

Combining all the factors obtained in the previous steps, we can express the polynomial in factor form:

64x^3 - y^3 - 8z^3 + 24xyz = 8(2x - y)(4x^2 + 2xy + y^2) - z^3 + 3xyz

Summary:

To express the polynomial 64x^3 - y^3 - 8z^3 + 24xyz in factor form, we first identified the common factor 8 and factored it out. Then we applied the difference of cubes formula to factor the remaining polynomial. Finally, we simplified the remaining terms and combined all the factors to obtain the final factor form.