f(x) = -3x² g(x) = -5x³ - 2x⁴ + 1 h(x) = 2 + 5x
1. add polynomials. f(x) + g(x)
2. subtract polynomials f(x) - h(x)
3. multiply f(x)*h(x)
4. determine the end behavior of the function f(x) and g(x) .
5. which calculation above will result in a polynomial degree 2? circle below.
f(x) + g(x) f(x) - h(x) f(x)*h(x) none of these
bonus write the product using the fewest number of terms.
(x + 3)(x² - 3x + 2)

Question

f(x) = -3x²  g(x) = -5x³ - 2x⁴ + 1  h(x) = 2 + 5x
1. add polynomials.  f(x) + g(x)
2. subtract polynomials  f(x) - h(x)
3. multiply  f(x)*h(x)
4. determine the end behavior of the function f(x) and g(x) .
5. which calculation above will result in a polynomial degree 2? circle below.
 f(x) + g(x)  f(x) - h(x)  f(x)*h(x)  none of these
bonus write the product using the fewest number of terms.
(x + 3)(x² - 3x + 2)

Accepted Answer

### 1. Add polynomials $ f(x) + g(x) $
# Explanation:
## Step1: Substitute the functions
$ f(x) = -3x^2 $, $ g(x) = -5x^3 - 2x^4 + 1 $, so $ f(x) + g(x)= -3x^2+(-5x^3 - 2x^4 + 1) $
## Step2: Combine like terms (no like terms here, just rearrange)
$ f(x) + g(x)= -2x^4 - 5x^3 - 3x^2 + 1 $
# Answer: $ -2x^4 - 5x^3 - 3x^2 + 1 $

### 2. Subtract polynomials $ f(x) - h(x) $
# Explanation:
## Step1: Substitute the functions
$ f(x) = -3x^2 $, $ h(x) = 2 + 5x $, so $ f(x)-h(x)= -3x^2-(2 + 5x) $
## Step2: Distribute the negative sign
$ f(x)-h(x)= -3x^2 - 2 - 5x $
## Step3: Rearrange terms
$ f(x)-h(x)= -3x^2 - 5x - 2 $
# Answer: $ -3x^2 - 5x - 2 $

### 3. Multiply polynomials $ f(x)\times h(x) $
# Explanation:
## Step1: Substitute the functions
$ f(x) = -3x^2 $, $ h(x) = 2 + 5x $, so $ f(x)\times h(x)= -3x^2\times(2 + 5x) $
## Step2: Distribute $ -3x^2 $
$ -3x^2\times2+(-3x^2)\times5x=-6x^2 - 15x^3 $
## Step3: Rearrange terms
$ f(x)\times h(x)= -15x^3 - 6x^2 $
# Answer: $ -15x^3 - 6x^2 $

### 4. Determine the end - behavior of $ f(x) $ and $ g(x) $
#### For $ f(x)=-3x^2 $
The leading term is $ -3x^2 $, the degree $ n = 2 $ (even) and the leading coefficient $ a=-3$ (negative).
For a polynomial $ y = a_nx^n+\cdots+a_0 $, if $ n $ is even and $ a_n<0 $, as $ x
ightarrow+\infty $, $ y
ightarrow-\infty $; as $ x
ightarrow-\infty $, $ y
ightarrow-\infty $.

#### For $ g(x)=-2x^4 - 5x^3 - 3x^2 + 1 $
The leading term is $ -2x^4 $, the degree $ n = 4 $ (even) and the leading coefficient $ a = - 2$ (negative).
If $ n $ is even and $ a_n<0 $, as $ x
ightarrow+\infty $, $ y
ightarrow-\infty $; as $ x
ightarrow-\infty $, $ y
ightarrow-\infty $.

# Answer:
- For $ f(x)=-3x^2 $: As $ x
ightarrow\pm\infty $, $ f(x)
ightarrow-\infty $
- For $ g(x)=-2x^4 - 5x^3 - 3x^2 + 1 $: As $ x
ightarrow\pm\infty $, $ g(x)
ightarrow-\infty $

### 5. Which calculation results in degree 2?
- For $ f(x)+g(x)= -2x^4 - 5x^3 - 3x^2 + 1 $, the degree is 4.
- For $ f(x)-h(x)= -3x^2 - 5x - 2 $, the degree is 2.
- For $ f(x)\times h(x)= -15x^3 - 6x^2 $, the degree is 3.

# Answer: $ f(x)-h(x) $

### BONUS: Expand $ (x + 3)(x^2-3x + 2) $
# Explanation:
## Step1: Distribute $ x $ and 3
$ x(x^2-3x + 2)+3(x^2-3x + 2) $
## Step2: Multiply each term
$ x\times x^2-x\times3x + x\times2+3\times x^2-3\times3x + 3\times2 $
$ =x^3-3x^2 + 2x+3x^2-9x + 6 $
## Step3: Combine like terms
$ x^3+(-3x^2 + 3x^2)+(2x-9x)+6 $
$ =x^3-7x + 6 $
# Answer: $ x^3-7x + 6 $

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QUESTION IMAGE

Question

Explanation:

1. Add polynomials \( f(x) + g(x) \)

Step1: Substitute the functions

Step2: Combine like terms (no like terms here, just rearrange)

Step1: Substitute the functions

Step2: Distribute the negative sign

Step3: Rearrange terms

Step1: Substitute the functions

Step2: Distribute \( -3x^2 \)

Step3: Rearrange terms

Answer:

2. Subtract polynomials \( f(x) - h(x) \)