Rectangular prism A has a volume of \begin{align*}x^3 +2x^2 - 3\end{align*}. Rectangular prism B has a volume of \begin{align*}x^4 + 2x^3 - 8x^2\end{align*}. How much larger is the volume of rectangular prism B than rectangular prism A?

### Adding and Subtracting Polynomials

A **polynomial** is an expression with multiple variable terms, such that the exponents are greater than or equal to zero. All quadratic and linear equations are polynomials. Equations with negative exponents, square roots, or variables in the denominator are not polynomials.

Now that we have established what a polynomial is, there are a few important parts. Just like with a quadratic, a polynomial can have a **constant**, which is a number without a variable. The **degree** of a polynomial is the largest exponent. For example, all quadratic equations have a degree of 2. Lastly, the **leading coefficient** is the coefficient in front of the variable with the degree. In the polynomial \begin{align*}4x^4 + 5x^3 -8x^2 + 12x +24\end{align*} above, the degree is 4 and the leading coefficient is also 4. Make sure that when finding the degree and leading coefficient you have the polynomial in standard form. **Standard form** lists all the variables in order, from greatest to least.

#### Solve the following problems

Rewrite \begin{align*}x^3 -5x^2 + 12x^4 + 15 -8x\end{align*} in standard form and find the degree and leading coefficient.

To rewrite in standard form, put each term in order, from greatest to least, according to the exponent. Always write the constant last.

\begin{align*}x^3 -5x^2 + 12x^4 + 15 -8x \rightarrow 12x^4 + x^3 -5x^2 -8x + 15\end{align*}

Now, it is easy to see the leading coefficient, 12, and the degree, 4.

Simplify \begin{align*}(4x^3 -2x^2 + 4x + 15) + (x^4 -8x^3 -9x-6)\end{align*}

To add or subtract two polynomials, combine like terms. **Like terms** are any terms where the exponents of the variable are the same. We will regroup the polynomial to show the like terms.

\begin{align*}& (4x^3 -2x^2 + 4x + 15) + (x^4 -8x^3 -9x -6)\\ & x^4 + (4x^3 -8x^3) -2x^2 + (4x -9x) + (15-6)\\ & x^4 -4x^3 -2x^2 -5x + 9\end{align*}

Simplify \begin{align*}(2x^3 + x^2 -6x -7)-(5x^3 -3x^2 + 10x -12)\end{align*}

When subtracting, distribute the negative sign to every term in the second polynomial, then combine like terms.

\begin{align*}& (2x^3 + x^2 -6x -7)-(5x^3 -3x^2 +10x -12)\\ & 2x^3 + x^2 -6x -7 -5x^3 + 3x^2 -10x + 12\\ & (2x^3 -5x^3) + (x^2 + 3x^2) + (-6x -10x) + (-7 + 12)\\ & -3x^3 + 4x^2 -16x + 5\end{align*}

### Examples

#### Example 1

Earlier, you were asked how much larger is the volume of the rectangular prism B than rectangular prism A.

We need to subtract the volume of rectangular prism A from the volume of rectangular prism B.

\begin{align*}(x^4 + 2x^3 - 8x^2) - (x^3 + 2x^2 - 3)\\ = x^4 + 2x^3 - 8x^2 - x^3 - 2x^2 + 3 \\ = x^4 + x^3 - 10x^2 + 3\end{align*}.

Therefore, the difference between the two volumes is \begin{align*}x^4 + x^3 - 10x^2 + 3\end{align*}.

#### Example 2

Is \begin{align*}\sqrt{2x^3 -5x} + 6\end{align*} a polynomial? Why or why not?

No, this is not a polynomial because \begin{align*}x\end{align*} is under a square root in the equation.

#### Example 3

Find the leading coefficient and degree of \begin{align*}6x^2 -3x^5 + 16x^4 + 10x -24\end{align*}.

In standard form, this polynomial is \begin{align*}-3x^5 + 16x^4 + 6x^2 + 10x -24\end{align*}. Therefore, the degree is 5 and the leading coefficient is -3.

#### Example 4

\begin{align*}(9x^2 + 4x^3 -15x + 22)+(6x^3 -4x^2 + 8x -14)\end{align*}

\begin{align*}(9x^2 + 4x^3 -15x + 22)+(6x^3 -4x^2 + 8x -14) = 10x^3 + 5x^2 -7x + 8\end{align*}

#### Example 5

\begin{align*}(7x^3 + 20x -3)-(x^3 -2x^2 + 14x -18)\end{align*}

\begin{align*}(7x^3 + 20x -3)-(x^3 -2x^2 + 14x -18) = 6x^3 + 2x^2 + 6x + 15\end{align*}

### Review

Determine if the following expressions are polynomials. If not, state why. If so, write in standard form and find the degree and leading coefficient.

- \begin{align*}\frac{1}{x^2} + x + 5\end{align*}
- \begin{align*}x^3 + 8x^4 -15x + 14x^2 -20\end{align*}
- \begin{align*}x^3 + 8\end{align*}
- \begin{align*}5x^{-2} + 9x^{-1} + 16\end{align*}
- \begin{align*}x^2 \sqrt{2} - x\sqrt{6} + 10\end{align*}
- \begin{align*}\frac{x^4 + 8x^2 +12}{3}\end{align*}
- \begin{align*}\frac{x^2-4}{x}\end{align*}
- \begin{align*}-6x^3 + 7x^5 -10x^6 + 19x^2 -3x +41\end{align*}

Add or subtract the following polynomials.

- \begin{align*}(x^3 + 8x^2 -15x + 11) + (3x^3 -5x^2 -4x + 9)\end{align*}
- \begin{align*}(-2x^4 + x^3 + 12x^2 + 6x -18)-(4x^4 -7x^3 + 14x^2 + 18x -25)\end{align*}
- \begin{align*}(10x^3 -x^2 + 6x + 3) + (x^4 -3x^3 + 8x^2 -9x + 16)\end{align*}
- \begin{align*}(7x^3 -2x^2 + 4x -5)-(6x^4 + 10x^3 + x^2 + 4x -1)\end{align*}
- \begin{align*}(15x^2 + x -27) + (3x^3 -12x + 16)\end{align*}
- \begin{align*}(2x^5 -3x^4 + 21x^2 + 11x -32)-(x^4 -3x^3 -9x^2 + 14x -15)\end{align*}
- \begin{align*}(8x^3 -13x^2 + 24)-(x^3 + 4x^2 -2x + 17) + (5x^2 + 18x -19)\end{align*}

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 6.4.