Adding, subtracting, and multiplying polynomials.

Adding, subtracting, and multiplying polynomials.

Adding and subtracting

• In polynomials, there is at least 3 or more variables, such as x, x², x³, etc.. Sometimes a number has no variable, hence in this case it's variable is "1".

• Each variable has a coefficient paired with it.

• In 4x³, the coefficient is "4" while the variable is "x³"

Adding polynomials

• When adding two polynomials, you can only add/subtract two terms when they have the same variables.

• Let's look at an example!

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Example

Let  $f\left(x\right)\ =\ x^{2\ }+\ 2x\ +\ 3$  and  $g\left(x\right)\ =\ 2x^{2\ }-3x\ +4$  

f(x) + g(x)
 $\rightarrow$   $\left(x^{2\ }+\ 2x\ +\ 3\right)\ +\ \left(2x^2-3x+4\right)$  
 $\rightarrow\ x^{2\ }+2x^2+2x-3x+3+4$  
 $\rightarrow3x^2-x+7$  

Subtracting polynomials

• The same rules apply for subtracting, only we do not add!

• If a polynomial is being subtracted, you simply subtract the coefficient. If the coefficient is negative, then we must change the sign to positive.

• Let's do an example!

Example

Let  $f\left(x\right)\ =\ x^{2\ }+\ 2x\ +\ 3$  and  $g\left(x\right)\ =\ 2x^{2\ }-3x\ +4$  

f(x) - g(x)
 $\rightarrow$   $\left(x^{2\ }+\ 2x\ +\ 3\right)\ -\ \left(2x^2-3x+4\right)$  
Distribute that negative to each coefficient! 
 $\rightarrow\left(x^{2\ }+2x+3\right)+\left(-2x^2+3x-4\right)$  
 $\rightarrow\ x^{2\ }-2x^2+2x+3x+3-4$  
 $\rightarrow-x^2+5x-1$  

Multiple Choice

Let

 $f\left(x\right)\ =\ 5x^{2\ }+2x\ -7\ and\ g\left(x\right)=-x^{2\ }+3x\ +2$  

 $f\left(x\right)\ +\ g\left(x\right)\ =?$  

•  $6x^{2\ }+5x\ +9$  

•  $4x^{2\ }+5x\ -5$  

•  $4x^{2\ }-x-9$  

•  $6x^2-x-9$  

Multiple Choice

Let

 $f\left(x\right)\ =\ 3x^{2\ }-5x\ +8\ and\ g\left(x\right)=2x^{2\ }+7x\ -1$  

 $f\left(x\right)\ -\ g\left(x\right)\ =?$  

•  $5x^{2\ }+10x\ +9$  

•  $5x^{2\ }-12x\ -7$  

•  $x^{2\ }-12x\ +9$  

•  $x^2+2x+7$  

Multiplying polynomials

• When multiplying two polynomials together, you must multiply each term of one of them with the terms of the other!

• So let us have the polynomial

   $f\left(x\right)\ =\ x^{2\ }+x\ +1\$  and we want multiply it by $g\left(x\right)\ =\ 2x^2\ +x\ +\ 3$  

•  $x^2,\ x,\ and\ 1\$ will multiply each term in g(x), and then you finally add each common term! 

Let us do an example!

Let  $f\left(x\right)\ =\ x^{2\ }+\ x\ +\ 1$  and  $g\left(x\right)\ =\ 2x^{2\ }+x\ +3$  like previously.

 $f\left(x\right)\times g\left(x\right)$  
 $\rightarrow$   $\left(x^{2\ }+\ x\ +\ 1\right)\ \times\ \left(2x^2+x+3\right)$  
Distribute each g(x) to each term of f(x)
(This can work with each polynomials roles switched, up to you!)
 $\rightarrow x^{2\ }\left(2x^{2\ }+x+3\right)+x\left(2x^2+x+3\right)+1\left(2x^2+x+3\right)$  
 $\rightarrow\ \left(2x^4+x^3+3x^2\right)+\left(2x^3+x^2+3x\right)+\left(2x^2+x+3\right)$  
 $\rightarrow\ \left(2x^4\right)+\left(x^3+2x^3\right)+\left(3x^2+x^2+2x^2\right)+\left(3x+x\right)+\left(3\right)$  
 $\rightarrow\ 2x^4+3x^3+6x^2+4x+3$  

Multiple Choice

Let

 $f\left(x\right)\ =\ x^{2\ }+3x\ +1\ and\ g\left(x\right)=x^{2\ }-2x\ -3$  

 $f\left(x\right)\ \times\ g\left(x\right)\ =?$  

•  $x^4+x^3+8x^2+11x+3$  

•  $x^4+x^3+x^2+x+1$  

•  $x^{4\ }+8x^3-12x^2\ +1$  

•  $x^4+x^3-8x^2-11x-3$