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Algebra 1

6.1.4 Adding and Subtracting Polynomial Functions

Algebra 16.1.4 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 ) = ___________

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

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

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

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 ) = ___________

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

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

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

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

Combining and Evaluating Polynomial Functions

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 )

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