Math  /  Algebra

Question8. Find (g+h)(x)(g+h)(x) if * 7 points g(x)=2x3+3x4h(x)=2x2x+3\begin{array}{l} g(x)=2 x^{3}+3 x-4 \\ h(x)=2 x^{2}-x+3 \end{array} 4x3+2x14x5+2x14 x^{3}+2 x-1 \quad 4 x^{5}+2 x-1 A B 2x3+2x2+2x18x612 x^{3}+2 x^{2}+2 x-1 \quad 8 x^{6}-1

Studdy Solution

STEP 1

What is this asking? We're adding two functions, g(x)g(x) and h(x)h(x), together! Watch out! Don't multiply the functions by accident, we're just adding them!
Also, make sure to combine like terms correctly.

STEP 2

1. Define the functions
2. Add the functions
3. Simplify the result

STEP 3

Alright, let's **define** our functions!
We have g(x)=2x3+3x4g(x) = 2x^3 + 3x - 4 and h(x)=2x2x+3h(x) = 2x^2 - x + 3.
Super important to keep track of those exponents!

STEP 4

Now, let's **add** those functions together!
We want to find (g+h)(x)(g+h)(x), which just means g(x)+h(x)g(x) + h(x).
So, we've got: (g+h)(x)=(2x3+3x4)+(2x2x+3)(g+h)(x) = (2x^3 + 3x - 4) + (2x^2 - x + 3)

STEP 5

Let's **group** our like terms together.
Remember, "like terms" are terms with the same variable raised to the same power.
It's like sorting apples with apples and oranges with oranges! (g+h)(x)=2x3+2x2+3xx4+3(g+h)(x) = 2x^3 + 2x^2 + 3x - x - 4 + 3

STEP 6

Time to **simplify**!
Combine those like terms.
We have 3xx=2x3x - x = 2x and 4+3=1-4 + 3 = -1.
So, our **final expression** becomes: (g+h)(x)=2x3+2x2+2x1(g+h)(x) = 2x^3 + 2x^2 + 2x - 1

STEP 7

Our final answer is 2x3+2x2+2x12x^3 + 2x^2 + 2x - 1, which is answer C!

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