Question

Add, subtract, or multiply, as indicated. Express your answer as a single polynomial in standard form. $$ \left(x^{2}-3 x-4\right)-\left(x^{3}-3 x^{2}+x+5\right) $$

Short answer

-x^3 + 4x^2 - 4x - 9

Step-by-step solution

- Distribute the negative sign

First, distribute the negative sign to each term in the second polynomial to change the signs of each term inside the parentheses.\[(x^2 - 3x - 4) - (x^3 - 3x^2 + x + 5) = (x^2 - 3x - 4) - x^3 + 3x^2 - x - 5\]

- Combine like terms

Next, group and combine the like terms from the polynomials.Collect the terms by their powers of x: constants, x, x^2, and x^3.\[ -x^3 + (x^2 + 3x^2) + (-3x - x) + (-4 - 5) \]Simplify each group of like terms: \[-x^3 + 4x^2 - 4x - 9\]

- Write the polynomial in standard form

Finally, ensure the polynomial is written in standard form, which orders the terms from the highest degree to the lowest degree.\[-x^3 + 4x^2 - 4x - 9\]

Key concepts

Polynomial Subtraction

Polynomial subtraction involves subtracting one polynomial from another.
To do this, we distribute the negative sign to each term in the polynomial being subtracted.
It's crucial to change the signs of all terms in the second polynomial.
For instance, consider subtracting \((x^{3} - 3x^{2} + x + 5) \) from \((x^{2} - 3x - 4) \).
After distributing the negative sign, it becomes:
\[(x^{2} - 3x - 4) - (x^{3} - 3x^{2} + x + 5) = (x^{2} - 3x - 4) - x^{3} + 3x^{2} - x - 5\].
Now, the problem becomes an addition of the polynomials with opposite signs for the second polynomial’s terms.

Combining Like Terms

Combining like terms means grouping terms that have the same variable raised to the same power.
After distributing the negative sign in polynomial subtraction, the next step is to combine like terms.
For the given exercise, collect the terms by their powers of x: constants, x, x², and x³.
Our expression is now: \[-x^3 + (x^2 + 3x^2) + (-3x - x) + (-4 - 5)\].
Combine the terms:
\[ -x^3 + 4x^2 - 4x - 9\].
This process makes it easier to simplify and solve polynomial equations.

Polynomial Standard Form

A polynomial is in standard form when its terms are ordered from the highest degree to the lowest degree.
Degrees are the powers of the variable, like x³ (degree 3), x² (degree 2), x (degree 1), and constants (degree 0).
For the given problem, the simplified polynomial is, \[-x^3 + 4x^2 - 4x - 9\].
This polynomial is already in standard form since the terms are ordered from the highest degree to the lowest degree.
Properly ordering terms helps in identifying the leading term, which is crucial for many algebraic operations and understanding polynomial behavior.