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Adding and Subtracting Polynomial Functions

6.1.4 Adding and Subtracting Polynomial Functions

6.1.4 • Adding and Subtracting Polynomial Functions

Activity

Add the polynomial functions. Evaluate the sum for each given value.

z ( x ) = 8 x 2 + 4 x − 6 z ( x ) = 8 x 2 + 4 x − 6

w ( x ) = − 5 x 2 + 2 x + 3 w ( x ) = − 5 x 2 + 2 x + 3

1. ( z + w ) ( x ) = ( z + w ) ( x ) = ___________

Compare your answer:

( z + w ) ( x ) = 3 x 2 + 6 x − 3 ( z + w ) ( x ) = 3 x 2 + 6 x − 3

2. ( z + w ) ( 6 ) = ( z + w ) ( 6 ) = ___________

Compare your answer:

( z + w ) ( 6 ) = 141 ( z + w ) ( 6 ) = 141

3. ( z + w ) ( − 9 ) = ( z + w ) ( − 9 ) = ___________

Compare your answer:

( z + w ) ( − 9 ) = 186 ( z + w ) ( − 9 ) = 186

4. ( z + w ) ( 0 ) ( z + w ) ( 0 ) ___________

Compare your answer:

( z + w ) ( 0 ) = − 3 ( z + w ) ( 0 ) = − 3

Subtract the polynomial functions. Evaluate the sum for each given value.

m ( x ) = − 6 x 2 + 2 x − 8 m ( x ) = − 6 x 2 + 2 x − 8

n ( x ) = − x 2 − 7 x − 6 n ( x ) = − x 2 − 7 x − 6

5. ( m − n ) ( x ) = ( m − n ) ( x ) = ___________

Compare your answer:

( m − n ) ( x ) = − 5 x 2 + 9 x − 2 ( m − n ) ( x ) = − 5 x 2 + 9 x − 2

6. ( m − n ) ( − 2 ) = ( m − n ) ( − 2 ) = ___________

Compare your answer:

( m − n ) ( − 2 ) = − 40 ( m − n ) ( − 2 ) = − 40

7. ( m − n ) ( 0 ) = ( m − n ) ( 0 ) = ___________

Compare your answer:

( m − n ) ( 0 ) = − 2 ( m − n ) ( 0 ) = − 2

8. ( m − n ) ( 4 ) = ( m − n ) ( 4 ) = ___________

Compare your answer:

( m − n ) ( 4 ) = − 46 ( m − n ) ( 4 ) = − 46

9. What is a good way to check that you found the correct sum or difference when adding or subtracting polynomial functions?

Compare your answer:

Substitute a value for x x into the individual polynomial functions. Add or subtract the results. The answer should be equal to the value you find when you substitute it into the sum or difference of the polynomial functions.

Video: Combining and Evaluating Polynomial Functions

Watch the following video to learn more about how to add and subtract polynomial functions and evaluate them at a given value:

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Self Check

For functions f ( x ) = 5 x 2 − 4 x − 1 and g ( x ) = x 2 + 3 x + 8 , find:

         ( f + g ) ( x ) = _______ 

         ( f + g ) ( 5 ) = _______

  1. ( f + g ) ( x ) = 6 x 2 − x + 7

    ( f + g ) ( 5 ) = 32

  2. ( f + g ) ( x ) = 4 x 2 − 7 x − 9

    ( f + g ) ( 5 ) = 56

  3. ( f + g ) ( x ) = 6 x 2 − x + 7

    ( f + g ) ( 5 ) = 152

  4. ( f + g ) ( x ) = 4 x 2 − x + 7

    ( f + g ) ( 5 ) = 102

Additional Resources

Adding and Subtracting Polynomial Functions

Just as polynomials can be added and subtracted, polynomial functions can also be added and subtracted.

For functions f ( x ) f ( x ) and g ( x ) g ( x ) ,

  • ( f + g ) ( x ) = f ( x ) + g ( x ) ( f + g ) ( x ) = f ( x ) + g ( x )
  • ( f − g ) ( x ) = f ( x ) − g ( x ) ( f − g ) ( x ) = f ( x ) − g ( x )

Let’s look at some examples.

For the functions f ( x ) = 3 x 2 − 5 x + 7 f ( x ) = 3 x 2 − 5 x + 7 and g ( x ) = x 2 − 4 x − 3 g ( x ) = x 2 − 4 x − 3 , find:

  1. ( f + g ) ( x ) ( f + g ) ( x )
  2. ( f + g ) ( 3 ) ( f + g ) ( 3 )
  3. ( f − g ) ( x ) ( f − g ) ( x )
  4. ( f − g ) ( − 2 ) ( f − g ) ( − 2 )

The following is a step-by-step breakdown.

a. ( f + g ) ( x ) = f ( x ) + g ( x ) f + g ) ( x ) = f ( x ) + g ( x )

Step 1 - Substitute

f ( x ) = 3 x 2 − 5 x + 7 f ( x ) = 3 x 2 − 5 x + 7 and g ( x ) = x 2 − 4 x − 3 g ( x ) = x 2 − 4 x − 3 .

( f + g ) ( x ) = ( 3 x 2 − 5 x + 7 ) + ( x 2 − 4 x − 3 ) ( f + g ) ( x ) = ( 3 x 2 − 5 x + 7 ) + ( x 2 − 4 x − 3 )

Step 2 - Rewrite without the parentheses.

( f + g ) ( x ) = 3 x 2 − 5 x + 7 − x 2 − 4 x − 3 ( f + g ) ( x ) = 3 x 2 − 5 x + 7 − x 2 − 4 x − 3

Step 3 - Put like terms together.

( f + g ) ( x ) = 3 x 2 + x 2 − 5 x − 4 x + 7 − 3 ( f + g ) ( x ) = 3 x 2 + x 2 − 5 x − 4 x + 7 − 3

Step 4 - Combine like terms.

( f + g ) ( x ) = 4 x 2 − 9 x + 4 ( f + g ) ( x ) = 4 x 2 − 9 x + 4

In part (a), we found ( f + g ) ( x ) ( f + g ) ( x ) , and now are asked to find ( f + g ) ( 3 ) ( f + g ) ( 3 ) .

b. ( f + g ) ( x ) = 4 x 2 − 9 x + 4 ( f + g ) ( x ) = 4 x 2 − 9 x + 4

Step 1 - To find ( f + g ) ( 3 ) ( f + g ) ( 3 ) , substitute x = 3 x = 3 .

( f + g ) ( 3 ) = 4 ( 3 ) 2 − 9 · 3 + 4 ( f + g ) ( 3 ) = 4 ( 3 ) 2 − 9 · 3 + 4

Step 2 - Simplify.

( f + g ) ( 3 ) = 4 · 9 − 9 · 3 + 4 ( f + g ) ( 3 ) = 4 · 9 − 9 · 3 + 4

( f + g ) ( 3 ) = 36 − 27 + 4 ( f + g ) ( 3 ) = 36 − 27 + 4

( f + g ) ( 3 ) = 13 ( f + g ) ( 3 ) = 13

Notice that we could have found ( f + g ) ( 3 ) ( f + g ) ( 3 ) by first finding the values of f ( 3 ) f ( 3 ) and g ( 3 ) g ( 3 ) separately and then adding the results.

Step 1 - Find f ( 3 ) f ( 3 ) .

f ( x ) = 3 x 2 − 5 x + 7 f ( x ) = 3 x 2 − 5 x + 7

f ( 3 ) = 3 ( 3 ) 2 − 5 ( 3 ) + 7 f ( 3 ) = 3 ( 3 ) 2 − 5 ( 3 ) + 7

f ( 3 ) = 19 f ( 3 ) = 19

Step 2 - Find g ( 3 ) g ( 3 ) .

g ( x ) = x 2 − 4 x − 3 g ( x ) = x 2 − 4 x − 3

g ( 3 ) = − 6 g ( 3 ) = − 6

Step 3 - Find ( f + g ) ( 3 ) ( f + g ) ( 3 ) .

( f + g ) ( x ) = f ( x ) + g ( x ) ( f + g ) ( x ) = f ( x ) + g ( x )

( f + g ) ( 3 ) = f ( 3 ) + g ( 3 ) ( f + g ) ( 3 ) = f ( 3 ) + g ( 3 )

Step 4 - Substitute f ( 3 ) = 1 9 f ( 3 ) = 1 9 and g ( 3 ) = − 6 g ( 3 ) = − 6 .

( f + g ) ( 3 ) = 19 + ( − 6 ) ( f + g ) ( 3 ) = 19 + ( − 6 )

( f + g ) ( 3 ) = 13 ( f + g ) ( 3 ) = 13

c. ( f − g ) ( x ) = f ( x ) − g ( x ) ( f − g ) ( x ) = f ( x ) − g ( x )

Step 1 - Substitute

f ( x ) = 3 x 2 − 5 x + 7 f ( x ) = 3 x 2 − 5 x + 7 and g ( x ) = x 2 − 4 x − 3 g ( x ) = x 2 − 4 x − 3 .

( f − g ) ( x ) = ( 3 x 2 − 5 x + 7 ) − ( x 2 − 4 x − 3 ) ( f − g ) ( x ) = ( 3 x 2 − 5 x + 7 ) − ( x 2 − 4 x − 3 )

Step 2 - Rewrite without the parentheses.

( f − g ) ( x ) = 3 x 2 − 5 x + 7 − x 2 + 4 x + 3 ( f − g ) ( x ) = 3 x 2 − 5 x + 7 − x 2 + 4 x + 3

Step 3 - Put like terms together.

( f − g ) ( x ) = 3 x 2 − x 2 − 5 x + 4 x + 7 + 3 ( f − g ) ( x ) = 3 x 2 − x 2 − 5 x + 4 x + 7 + 3

Step 4 - Combine like terms.

( f − g ) ( x ) = 2 x 2 − x + 10 ( f − g ) ( x ) = 2 x 2 − x + 10

d. ( f − g ) ( x ) = 2 x 2 − x + 10 ( f − g ) ( x ) = 2 x 2 − x + 10

Step 1 - To find ( f − g ) ( − 2 ) ( f − g ) ( − 2 ) , substitute x = − 2 x = − 2 .

( f − g ) ( − 2 ) = 2 ( − 2 ) 2 − ( − 2 ) + 10 ( f − g ) ( − 2 ) = 2 ( − 2 ) 2 − ( − 2 ) + 10

Step 2 - Simplify.

( f − g ) ( − 2 ) = 2 · 5 − ( − 2 ) + 10 ( f − g ) ( − 2 ) = 2 · 5 − ( − 2 ) + 10

( f − g ) ( − 2 ) = 20 ( f − g ) ( − 2 ) = 20

Adding and Subtracting Polynomial Functions

For functions f ( x ) = 2 x 2 − 4 x + 3 f ( x ) = 2 x 2 − 4 x + 3 and g ( x ) = x 2 − 2 x − 6 g ( x ) = x 2 − 2 x − 6 , find:

  1. ( f + g ) ( x ) ( f + g ) ( x )
  2. ( f + g ) ( 3 ) ( f + g ) ( 3 )
  3. ( f − g ) ( x ) ( f − g ) ( x )
  4. ( f − g ) ( − 2 ) ( f − g ) ( − 2 )

Here is how to add and subtract the polynomial functions:

  1. ( f + g ) ( x ) = ( 2 x 2 − 4 x + 3 ) + ( x 2 − 2 x − 6 ) = 2 x 2 − 4 x + 3 + x 2 − 2 x − 6 = 3 x 2 − 6 x − 3 ( f + g ) ( x ) = ( 2 x 2 − 4 x + 3 ) + ( x 2 − 2 x − 6 ) = 2 x 2 − 4 x + 3 + x 2 − 2 x − 6 = 3 x 2 − 6 x − 3
  2. ( f + g ) ( 3 ) = 3 ( 3 ) 2 − 6 ( 3 ) − 3 = 3 ( 9 ) − 6 ( 3 ) − 3 = 27 − 18 − 3 = 6 ( f + g ) ( 3 ) = 3 ( 3 ) 2 − 6 ( 3 ) − 3 = 3 ( 9 ) − 6 ( 3 ) − 3 = 27 − 18 − 3 = 6
  3. ( f − g ) ( x ) = ( 2 x 2 − 4 x + 3 ) − ( x 2 − 2 x − 6 ) = 2 x 2 − 4 x + 3 − x 2 + 2 x + 6 = x 2 − 2 x + 9 ( f − g ) ( x ) = ( 2 x 2 − 4 x + 3 ) − ( x 2 − 2 x − 6 ) = 2 x 2 − 4 x + 3 − x 2 + 2 x + 6 = x 2 − 2 x + 9
  4. ( f − g ) ( − 2 ) = ( − 2 ) 2 − 2 ( − 2 ) + 9 = ( 4 ) + 4 + 9 = 17 ( f − g ) ( − 2 ) = ( − 2 ) 2 − 2 ( − 2 ) + 9 = ( 4 ) + 4 + 9 = 17