Add And Subtract Polynomial Functions - Worksheet - Free Printable

Let's solve each problem step by step. We are asked to add and subtract polynomial functions, which means we combine like terms (terms with the same variable and exponent) either by adding or subtracting them.

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1. Find $ (f + g)(y) $

Given:
- $ f(y) = -7y^6 - y^2 - 11 $
- $ g(y) = 8y^6 - 2y^2 + 17 $

Add the two functions:

$$
(f + g)(y) = f(y) + g(y)
= (-7y^6 + 8y^6) + (-y^2 - 2y^2) + (-11 + 17)
= (1y^6) + (-3y^2) + (6)
$$

✔ Answer: $ (f + g)(y) = y^6 - 3y^2 + 6 $

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2. Find $ (f - g)(x) $

Given:
- $ f(x) = -11x^5 - 9x^4 $
- $ g(x) = 18x^5 - 4x^4 $

Subtract $ g(x) $ from $ f(x) $:

$$
(f - g)(x) = f(x) - g(x)
= (-11x^5 - 18x^5) + (-9x^4 + 4x^4)
= (-29x^5) + (-5x^4)
$$

✔ Answer: $ (f - g)(x) = -29x^5 - 5x^4 $

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3. Find $ (f + g)(a) $

Given:
- $ f(a) = -8a^{13} - 15a^{10} - 17a^8 - 19a^7 $
- $ g(a) = -5a^{13} + 6a^{10} + 20a^8 + 13a^7 $

Add:

$$
(f + g)(a) = (-8a^{13} - 5a^{13}) + (-15a^{10} + 6a^{10}) + (-17a^8 + 20a^8) + (-19a^7 + 13a^7)
= (-13a^{13}) + (-9a^{10}) + (3a^8) + (-6a^7)
$$

✔ Answer: $ (f + g)(a) = -13a^{13} - 9a^{10} + 3a^8 - 6a^7 $

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4. Find $ (f - g)(b) $

Given:
- $ f(b) = -8b^{13} - 6b $
- $ g(b) = 4b^{13} - 4b $

Subtract:

$$
(f - g)(b) = f(b) - g(b)
= (-8b^{13} - 4b^{13}) + (-6b + 4b)
= (-12b^{13}) + (-2b)
$$

✔ Answer: $ (f - g)(b) = -12b^{13} - 2b $

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5. Find $ (f + g)(k) $

Given:
- $ f(k) = -19k^{12} - 16k^9 - 10k^8 - 4k^5 $
- $ g(k) = 9k^{12} + 5k^9 + 20k^8 + 7k^5 $

Add:

$$
(f + g)(k) = (-19k^{12} + 9k^{12}) + (-16k^9 + 5k^9) + (-10k^8 + 20k^8) + (-4k^5 + 7k^5)
= (-10k^{12}) + (-11k^9) + (10k^8) + (3k^5)
$$

✔ Answer: $ (f + g)(k) = -10k^{12} - 11k^9 + 10k^8 + 3k^5 $

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6. Find $ (f - g)(6) $

We are to evaluate $ (f - g)(6) $, so first find $ f(6) $ and $ g(6) $, then subtract.

Given:
- $ f(k) = -4k^5 - 20k^4 - 6 $
- $ g(k) = 17k^5 - 18k^4 + 9 $

First compute $ f(6) $:

$$
f(6) = -4(6)^5 - 20(6)^4 - 6
= -4(7776) - 20(1296) - 6
= -31104 - 25920 - 6 = -57030
$$

Now $ g(6) $:

$$
g(6) = 17(6)^5 - 18(6)^4 + 9
= 17(7776) - 18(1296) + 9
= 132192 - 23328 + 9 = 108873
$$

Now subtract:

$$
(f - g)(6) = f(6) - g(6) = -57030 - 108873 = -165903
$$

✔ Answer: $ (f - g)(6) = -165903 $

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7. Find $ (f + g)(6) $

Given:
- $ f(n) = -17n^4 - 15n^2 $
- $ g(n) = 8n^4 - 20n^2 $

First add the functions:

$$
(f + g)(n) = (-17n^4 + 8n^4) + (-15n^2 - 20n^2) = -9n^4 - 35n^2
$$

Now evaluate at $ n = 6 $:

$$
(f + g)(6) = -9(6)^4 - 35(6)^2
= -9(1296) - 35(36)
= -11664 - 1260 = -12924
$$

✔ Answer: $ (f + g)(6) = -12924 $

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8. Find $ (f + g)(5) $

Given:
- $ f(z) = -12z - 5 $
- $ g(z) = 2z - 3 $

Add the functions:

$$
(f + g)(z) = (-12z + 2z) + (-5 - 3) = -10z - 8
$$

Now plug in $ z = 5 $:

$$
(f + g)(5) = -10(5) - 8 = -50 - 8 = -58
$$

✔ Answer: $ (f + g)(5) = -58 $

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✔ Final Answers Summary:

1. $ (f + g)(y) = y^6 - 3y^2 + 6 $
2. $ (f - g)(x) = -29x^5 - 5x^4 $
3. $ (f + g)(a) = -13a^{13} - 9a^{10} + 3a^8 - 6a^7 $
4. $ (f - g)(b) = -12b^{13} - 2b $
5. $ (f + g)(k) = -10k^{12} - 11k^9 + 10k^8 + 3k^5 $
6. $ (f - g)(6) = -165903 $
7. $ (f + g)(6) = -12924 $
8. $ (f + g)(5) = -58 $

Let me know if you'd like these written out neatly for printing or review!