6.4: Adding and Subtracting Polynomials
Rectangular prism A has a volume of \begin{align*}x^3 +2x^2  3\end{align*}
Watch This
James Sousa: Ex: Intro to Polynomials in One Variable
Guidance
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*}
Example A
Rewrite \begin{align*}x^3 5x^2 + 12x^4 + 15 8x\end{align*}
Solution: 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.
Example B
Simplify \begin{align*}(4x^3 2x^2 + 4x + 15) + (x^4 8x^3 9x6)\end{align*}
Solution: 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) + (156)\\
& x^4 4x^3 2x^2 5x + 9\end{align*}
Example C
Simplify \begin{align*}(2x^3 + x^2 6x 7)(5x^3 3x^2 + 10x 12)\end{align*}
Solution: 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*}
Intro Problem Revisit
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*}
Guided Practice
1. Is \begin{align*}\sqrt{2x^3 5x} + 6\end{align*}
2. Find the leading coefficient and degree of \begin{align*}6x^2 3x^5 + 16x^4 + 10x 24\end{align*}
Add or subtract.
3. \begin{align*}(9x^2 + 4x^3 15x + 22)+(6x^3 4x^2 + 8x 14)\end{align*}
4. \begin{align*}(7x^3 + 20x 3)(x^3 2x^2 + 14x 18)\end{align*}
Answers
1. No, this is not a polynomial because \begin{align*}x\end{align*}
2. In standard form, this polynomial is \begin{align*}3x^5 + 16x^4 + 6x^2 + 10x 24\end{align*}
3. \begin{align*}(9x^2 + 4x^3 15x + 22)+(6x^3 4x^2 + 8x 14) = 10x^3 + 5x^2 7x + 8\end{align*}
4. \begin{align*}(7x^3 + 20x 3)(x^3 2x^2 + 14x 18) = 6x^3 + 2x^2 + 6x + 15\end{align*}
Vocabulary
 Polynomial
 An expression with multiple variable terms, such that the exponents are greater than or equal to zero.
 Constant
 A number without a variable in a mathematical expression.
 Degree(of a polynomial)
 The largest exponent in a polynomial.
 Leading coefficient
 The coefficient in front of the variable with the degree.
 Standard form
 Lists all the variables in order, from greatest to least.
 Like terms
 Any terms where the exponents of the variable are the same.
Practice
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*}
1x2+x+5 
\begin{align*}x^3 + 8x^4 15x + 14x^2 20\end{align*}
x3+8x4−15x+14x2−20 
\begin{align*}x^3 + 8\end{align*}
x3+8 
\begin{align*}5x^{2} + 9x^{1} + 16\end{align*}
5x−2+9x−1+16 
\begin{align*}x^2 \sqrt{2}  x\sqrt{6} + 10\end{align*}
x22√−x6√+10 
\begin{align*}\frac{x^4 + 8x^2 +12}{3}\end{align*}
x4+8x2+123 
\begin{align*}\frac{x^24}{x}\end{align*}
x2−4x 
\begin{align*}6x^3 + 7x^5 10x^6 + 19x^2 3x +41\end{align*}
−6x3+7x5−10x6+19x2−3x+41
Add or subtract the following polynomials.

\begin{align*}(x^3 + 8x^2 15x + 11) + (3x^3 5x^2 4x + 9)\end{align*}
(x3+8x2−15x+11)+(3x3−5x2−4x+9) 
\begin{align*}(2x^4 + x^3 + 12x^2 + 6x 18)(4x^4 7x^3 + 14x^2 + 18x 25)\end{align*}
(−2x4+x3+12x2+6x−18)−(4x4−7x3+14x2+18x−25)  \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*}
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constant
A constant is a value that does not change. In Algebra, this is a number such as 3, 12, 342, etc., as opposed to a variable such as x, y or a.Degree
The degree of a polynomial is the largest exponent of the polynomial.distributive property
The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, .Leading coefficient
The leading coefficient of a polynomial is the coefficient of the variable with the highest degree.like terms
Terms are considered like terms if they are composed of the same variables with the same exponents on each variable.Polynomial
A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.Standard form
The standard form of a polynomial lists all the terms in order, from highest degree to lowest degree.Image Attributions
Here you'll learn how to add and subtract polynomials, as well as learn about the different parts of a polynomial.