2.2: Polynomial Addition, Subtraction, Multiplication and Monomial Division

Much of Section \(2.2\) is mathematics you already know, like combining like terms when adding polynomials, but we’ll keep you on your toes with the introduction of function notation! You’ll also get the added practice you crave for the various exponent properties. This section aims to strengthen your ability to work with polynomial functions using function notation.

Adding Functions f and g may be notated (f+g)(x)

\[(f+g)(x) = f(x) + g(x)\]

Example 2.2.1

\(f(x) = 10x^3 − 6x^2 + 5\) and \(g(x) = 1 − 4x^3\). Find \((f+g)(x)\).

Solution

\[ \begin{align*} (f+g)(x) &= \left( \underbrace{10x^3 − 6x^2 + 5}_{f(x)} \right) + \left( \underbrace{1 − 4x^3}_{g(x)} \right) \\[4pt] &= 10x^3 + (−6x^2 ) + 5 + 1 + (−4x^3) &\text{Turn subtraction into addition: \(A-B=A+(-B)\).} \\[4pt] &= 10x^3 + (−4x^3 ) + (−6x^2 ) + 5 + 1 &\text{Rearrange terms so like terms are side-by-side.} \\[4pt] &= 6x^3 − 6x^2 + 6 &\text{Simplify by combining like terms.} \end{align*}\]

Subtracting Functions f and g may be notated (f-g)(x)

\[(f-g)(x) = f(x) - g(x)\]

Caution

Subtracting functions must be handled with greater care than adding functions. Remember that the quantity \(g(x)\) must be subtracted from \(f(x)\). Place parentheses around \(g(x)\) and subtract the quantity.

Example 2.2.2

\(f(x) = 2x^2 − 8x − 4\) and \(g(x) = 3x − 9\). Find \((f − g)(x)\).

Solution

\((f-g)(x) = \left( \underbrace{2x^2 − 8x − 4}_{f(x)} \right) - \left( \underbrace{3x − 9}_{g(x)} \right)\)

\(\begin{array} &&\underbrace{2x^2 − 8x − 4}_{f(x)} - \underbrace{(3x +(− 9))}_{g(x)} &\text{Consider the sign value of each term.} \\ &= 2x^2 + (−8x) + (−4) − 3x − (−9) &\text{Subtract each term of \(g\). Then simplify.} \\ &= 2x^2 + (−8x) + (−4) − 3x + 9 &A - -(B) - A+B.\\ &= 2x^2 + (−8x) + (−3x) + (−4) + 9 &\text{Rearrange terms so like terms are side-by-side.} \\ &= 2x^2 − 11x + 5 &\text{Simplify.} \end{array}\)

Multiplying Functions f and g may be notated (f⋅g)(x)

\[(f \cdot g)(x) = f(x) \cdot g(x)\]

Example 2.2.3

\(f(x) = 6 − 2x\) and \(g(x) = 3x^2 − 8x − 4\). Find \((f \cdot g)(x)\).

Solution

\( (f \cdot g)(x) = \left( \underbrace{6 − 2x}_{f(x)} \right) - \left( \underbrace{3x^2 − 8x − 4}_{g(x)} \right) \)

The table organizes terms. It’s clear what terms must be multiplied. Organization means fewer mistakes!

	\(3x^2\)	\(-8x\)	\(-4\)
\(6\)	\(18x^2\)	\(-48x\)	\(-24\)
\(-2x\)	\(-6x^3\)	\(16x^2\)	\(8x\)

\(\begin{array} &&= −6x^3 + 18x^2 + 16x^2 − 48x + 8x − 24 &\text{Gather like terms. Then simplify.} \\ &= −6x^3 + 34x^2 − 40x + 8x &\end{array}\)

Multiplying Functions f and g may be notated (f/g)(x)

\[\left( \dfrac{f}{g} \right)(x) = \dfrac{f(x)}{g(x)}\]

Example 2.2.4

\(f(x) = 8x^4\) and \(g(x) = 2x^9\). Find \(\left( \dfrac{f}{g} \right)(x)\).

Solution

\(\begin{array} &&\left( \dfrac{f}{g} \right)(x) = \dfrac{8x^4}{2x^9} &\text{The numerator and denominator are each monomials.} \\ &\dfrac{8x^4}{2x^9} = \dfrac{8}{2} \cdot \dfrac{x^4}{x^9} = 4x^{4−9} = 4x −5 &\text{Use the Quotient of Powers Property to simplify.} \\ &\dfrac{4}{x^5} &\text{Answer with positive exponents.} \end{array}\)

Definition: Rational Function

A rational function is a function \(\dfrac{f(x)}{g(x)}\), where both \(f(x)\) and \(g(x)\) are polynomial functions and \(g(x)\) is not zero itself.

We will study rational function in more depth in Chapter \(5\).

Try It! (Exercises)

1. Let \(f(x) = 8x + 2\) and \(g(x) = 3 − 4x\). Find each of the following:

1. \((f+g)(x)\)
2. \((f-g)(x)\)
3. \((f \cdot g)(x)\)

2. Find \(\left( \dfrac{f}{g} \right)(x)\) given the functions \(f\) and \(g\) below.

1. \(f(x) = 10x^{20}\) and \(g(x) = 20x^8\)
2. \(f(x) = \dfrac{3}{4}x^5\) and \(g(x) = \dfrac{1}{8}x^7\)

3. Let \(f(x) = x^3 − 6x^2 + 2\) and \(g(x) = 5x^2 − 4\). Find each of the following:

1. \((f+g)(x)\)
2. \((f-g)(x)\)
3. \((f \cdot g)(x)\)

4. In general, Is \((f-g)(x)\) the same as \((g-f)(x)\)? Explain.

5a. The width of a rectangle is described by the function \(f(x) = 2x − 5\). The length of the rectangle is described by the function \(g(x) = 3x + 1\). Which of the following describes the area of the rectangle?

1. \((f+g)(x)\)
2. \((f-g)(x)\)
3. \((f \cdot g)(x)\)

5b. Find the area of the rectangle as a polynomial function.

6) Since \((f+g)(x) = f(x) + g(x)\), the notation also applies to specific values of \(x\). That is, \((f + g)(2) = f(2) + g(2)\). Evaluate each of the following given \(f(x) = −4x^2 + 2x\) and \(g(x) = 1 − x^2\).

1. \((f+g)(2)\)
2. \((f+g)(0)\)
3. \((f+g)(-1)\)
4. \((f+g)(-2)\)

7) Since \((f \cdot g)(x) =f(x) \cdot g(x)\), the notation also applies to specific values of \(x\). That is, \((f \cdot g)(2) = f(2) \cdot g(2)\). Evaluate each of the following given \(f(x) = −4x^2 + 2x\) and \(g(x) = 1 −x^2\).

1. \((f \cdot g)(2)\)
2. \((f\cdot g)(0)\)
3. \((f\cdot g)(-1)\)
4. \((f\cdot g)(-2)\)