Skill: Add and subtract polynomial functions

To add or subtract polynomial functions, combine like terms: terms with the same variable and exponent.

For functions $f$ and $g$:

$$(f+g)(x)=f(x)+g(x)$$ $$(f-g)(x)=f(x)-g(x)$$

When subtracting, distribute the negative sign to every term in the second polynomial.

Example

Let

$$f(x)=3x^3-2x^2+5x-1$$

and

$$g(x)=x^3+4x^2-2x+6.$$

Find $(f+g)(x)$ and $(f-g)(x)$.

1. Add the functions

Write the polynomials together and combine like terms:

$$\begin{aligned} (f+g)(x) &=(3x^3-2x^2+5x-1)+(x^3+4x^2-2x+6)\\ &=(3x^3+x^3)+(-2x^2+4x^2)+(5x-2x)+(-1+6)\\ &=4x^3+2x^2+3x+5. \end{aligned}$$

So,

$$\boxed{(f+g)(x)=4x^3+2x^2+3x+5}$$

2. Subtract the functions

Distribute the negative sign across every term in $g(x)$:

$$\begin{aligned} (f-g)(x) &=(3x^3-2x^2+5x-1)-(x^3+4x^2-2x+6)\\ &=3x^3-2x^2+5x-1-x^3-4x^2+2x-6. \end{aligned}$$

Now combine like terms:

$$\begin{aligned} (f-g)(x) &=(3x^3-x^3)+(-2x^2-4x^2)+(5x+2x)+(-1-6)\\ &=2x^3-6x^2+7x-7. \end{aligned}$$

Therefore,

$$\boxed{(f-g)(x)=2x^3-6x^2+7x-7}$$

Always check that you combine only terms with the same exponent, and remember that subtraction changes the sign of every term in the polynomial being subtracted.