Question

Add, subtract, or multiply, as indicated. Express your answer as a single polynomial in standard form. $$ 8\left(4 x^{3}-3 x^{2}-1\right)-6\left(4 x^{3}+8 x-2\right) $$

Short answer

8x^3 - 24x^2 - 48x + 4

Step-by-step solution

- Distribute the Constants

Distribute 8 to each term inside the first parentheses and distribute -6 to each term inside the second parentheses. 8(4x^3 - 3x^2 - 1) = 32x^3 - 24x^2 - 8 -6(4x^3 + 8x - 2) = -24x^3 - 48x + 12

- Combine Like Terms

Next, combine like terms by adding the respective coefficients. 32x^3 - 24x^2 - 8 + (-24x^3 - 48x + 12) Combining the like terms: (32x^3 - 24x^3) + (-24x^2) + (-8 + 12) + (-48x)

- Simplify the Expression

Finally, simplify the above expression by performing the addition and subtraction operations: 32x^3 - 24x^3 = 8x^3 -8 + 12 = 4 The expression are now: 8x^3 - 24x^2 - 48x + 4

Key concepts

distributing constants

Distributing constants is a crucial step in polynomial operations. When you are distributing a constant through a polynomial, you multiply the constant by each term inside the parentheses.

For example, let's distribute the constants in the following expression:
\[ 8(4 x^{3}-3 x^{2}-1) - 6(4 x^{3}+8 x-2) \]

First, distribute the 8 through the first set of parentheses:
\[ 8(4 x^{3}-3 x^{2}-1) = 32x^{3} - 24x^{2} - 8 \]
Next, distribute -6 through the second set of parentheses:
\[ -6(4 x^{3}+8 x-2) = -24x^{3} - 48x + 12 \]

Now that we have distributed the constants, we can move on to the next step.

combining like terms

Combining like terms simplifies our polynomial expression. Like terms are those terms that have the same variable part.

After distributing the constants in the equation:
\[ 32x^{3} - 24x^{2} - 8 - 24x^{3} - 48x + 12 \]
We combine the like terms:
\[ (32x^{3} - 24x^{3}) + (-24x^{2}) + (-8 + 12) + (-48x) \]
Each type of term has its own category:

- The \(x^{3}\) terms: \(32x^{3} - 24x^{3}\)
- The \(x^{2}\) term: \(-24x^{2}\)
- The \(x\) term: \(-48x\)
- The constant terms: \(-8 + 12\)

By arranging and combining these like terms, you help to simplify the polynomial.

simplifying polynomials

Simplifying polynomials means combining all like terms and making the polynomial as concise as possible. We combine like terms:
\[32x^{3} - 24x^{3} = 8x^{3} \] \[-8 + 12 = 4 \]

Then, we incorporate these results with the other terms:
\[8x^{3} - 24x^{2} - 48x + 4 \]

This final expression represents the simplified form of the initial given polynomial. By following each step carefully—distributing constants, combining like terms, and simplifying—you can solve polynomial operations efficiently.