Introduction

Suppose that we have the two polynomials 6x^2 + x +1 and 2x^2 - 3x + 6 , and we want to add them. To do that, we put a plus sign between the two, collect like terms, and then simplify:

\begin{align} (6x^2+ x +1) + ( 2x^2 - 3x +6) & = \\[5pt] 6x^2 +x +1 + 2x^2 - 3x +6& = \\[5pt] \underbrace{6x^2 + 2x^2}_{x^2\textrm{ terms}} + \overbrace{x - 3x}^{x \textrm{ terms}} +\underbrace{1+6}_{\textrm{constants}} &= \\[5pt] (6+2) x^2 + (1-3) x + 7 & = \\[5pt] 8x^2 - 2 x + 7 \end{align}

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Example: Adding Polynomials

Add the polynomials x^3-x^2 - 2 and -2x^3 + 3x^2-x.

EXPLANATION

We put a plus sign between the two polynomials, collect like terms, and then simplify:

\begin{align*} (x^3-x^2 - 2) + (-2x^3 + 3x^2-x) &=\\[5pt] x^3-x^2 - 2 -2x^3 + 3x^2-x &=\\[5pt] x^3 - 2x^3 - x^2 +3x^2 -x-2&=\\[5pt] (1-2)x^3 + (-1+3)x^2 -x -2 &=\\[5pt] (-1)x^3 + (2)x^2 - x - 2 &=\\[5pt] -x^3+2x^2-x-2& \end{align*}

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Practice: Adding Polynomials

Add the polynomials $3b^4+3b^2$ and $-3b^2-3b^3.$

a
 $3b^4+b^3-6b^2$
b
 $3b^4-3b^3+b^2$
c
 $b^4-3b^3$
d
 $3b^4-3b^3$
e
 $3b^4+6b^3$
Practice: Adding Polynomials

Write the following polynomial in standard form.

a
b
c
d
e
Practice: Adding Polynomials

Add the polynomials $3x^4-6x^2+9$ and $8x^2-4x+2.$

a
 $3x^4-14x^2-4x+11$
b
 $3x^4+2x^2-4x+11$
c
 $x^4-x^2-4x+11$
d
 $5x^2-4x+11$
e
 $3x^4+14x^2-4x+11$
Example: Adding Polynomials Expressed as Functions

If f(x) = 1-2x^3 and g(x)=1+3x^3, then what is f(x) + g(x)?

EXPLANATION

We put a plus sign between the two polynomials, collect like terms, and then simplify:

\begin{align*} f(x) + g(x) &=\\[5pt] (1-2x^3) + (1+3x^3) &=\\[5pt] -2x^3 + 3x^3 + 1 + 1 &=\\[5pt] (-2+3)x^3 + (1+1) &=\\[5pt] (1)x^3 + (2) &=\\[5pt] x^3 + 2 \end{align*}

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Practice: Adding Polynomials Expressed as Functions

Given that $f(y)=3y^2-2$ and $g(y)=5y+7,$ write $f(y)+ g(y)$ in standard form.

a
b
c
d
e
Practice: Adding Polynomials Expressed as Functions

If $f(b)=b^2+3b+4$, $g(b)= b^2-1,$ then $f(b)+g(b)=$

a
 $2b^2-3b+4$
b
 $b^2+3b+4$
c
 $b^2+4b+3$
d
 $3b+3$
e
 $2b^2+3b+3$
Practice: Adding Polynomials Expressed as Functions

Given that $f(x)=x^3+2x -3$ and $g(x)=-3x^3 -5x+3,$ write $f(x)+g(x)$ in standard form.

a
b
c
d
e
The Difference of Two Polynomials

Suppose that we want to subtract the polynomial 2x^2 + x from the polynomial 6x^2 - 3x +1. To do that, we start by putting a minus sign between the two:

( 6x^2 - 3x +1) - ( 2x^2 + x )

Here, we need to remember to distribute the minus sign. Once we've done that, we can collect like terms and then simplify as usual:

\begin{align} (6x^2 - 3x +1) - ( 2x^2 + x ) & = \\[5pt] 6x^2 - 3x +1 - 2x^2 - x & = \\[5pt] 6x^2 - 2x^2 - 3x - x +1 &= \\[5pt] (6-2) x^2 + (-3-1) x + 1 & = \\[5pt] 4x^2 - 4 x + 1 \end{align}

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Example: Subtracting Polynomials

Subtract the polynomial 1-x-2x^2 from x^2+x+6.

EXPLANATION

We put a minus sign between the two polynomials, distribute it over the second polynomial, collect like terms, and then simplify:

\begin{align*} (x^2+x+6) - (1-x-2x^2) &=\\[5pt] x^2+x+6 - 1 + x+2x^2 &=\\[5pt] (1+2)x^2 + (1+1)x + (6-1) &=\\[5pt] 3x^2 +2x +5& \end{align*}

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Practice: Subtracting Polynomials

Subtract $4+23x$ from $2x+3x.$

a
 $x-4$
b
 $22x-4$
c
 $22x$
d
 $-4$
e
 $-18x-4$
Practice: Subtracting Polynomials

Write the following polynomial in standard form.

a
b
c
d
e
Practice: Subtracting Polynomials

Subtract $x^2-9x+3$ from $5x^3+2x^2-5.$

a
 $-5x^3 - x^2 - 9x+4$
b
 $-5x^3 + 3x^2 - 9x-8$
c
 $5x^3 - 3x^2 + 9x-2$
d
 $5x^3 - x^2 + 9x-2$
e
 $5x^3 + x^2 + 9x-8$
Example: Subtracting Polynomials Expressed as Functions

If f(x) = x^2-6 and g(x) =4 - x^2, what is f(x)-g(x)?

EXPLANATION

We put a minus sign between the two polynomials, distribute the minus sign, collect like terms, and then simplify:

\begin{align} f(x)-g(x) &=\\[5pt] (x^2-6) - (4 - x^2)&=\\[5pt] x^2-6 - 4 + x^2&=\\[5pt] (x^2 + x^2) + (-6 - 4)&=\\[5pt] 2x^2-10& \end{align}

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Practice: Subtracting Polynomials Expressed as Functions

Given that $f(x)=3x^2+1$ and $g(x)= x^2+3x,$ write $f(x) - g(x)$ in standard form.

a
b
c
d
e
Practice: Subtracting Polynomials Expressed as Functions

Given that $f(x)=2x^4+5x$ and $g(x)= 3x^2-7x,$ write $f(x) - g(x)$ in standard form.

a
b
c
d
e
Practice: Subtracting Polynomials Expressed as Functions

If $f(u)=3u^4+2u^3+u-1$ and $g(u)= 3u^4-4u^3+2u-1,$ then $f(u)-g(u)=$

a
 $6u^4 + 6u^3 - u$
b
 $6u^3 - 3u$
c
 $6u^3 - u$
d
 $6u^3 - u + 2$
e
 $2u^3 - 3u$
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