Math  /  Algebra

QuestionConsider the following polynomial function. f(x)=x4+2x313x210x+40f(x)=x^{4}+2 x^{3}-13 x^{2}-10 x+40
Step 2 of 4 : Find the degree and the yy-intercept. Express the intercept as an ordered pair.

Studdy Solution

STEP 1

What is this asking? We need to find the highest power of xx in our function and where the function crosses the yy-axis. Watch out! Don't mix up the degree and the yy-intercept!
The degree is about the biggest power of xx, and the yy-intercept is where our graph hits the yy-axis!

STEP 2

1. Find the Degree
2. Find the y-intercept

STEP 3

Let's **look** at our function f(x)=x4+2x313x210x+40f(x) = x^4 + 2x^3 - 13x^2 - 10x + 40.
The **degree** of a polynomial is the highest power of xx.

STEP 4

In our function, the **highest power** is x4x^4.
That means the **degree** of our polynomial is **4**!

STEP 5

The **yy-intercept** is the point where the graph of our function crosses the yy-axis.
This happens when x=0x = 0.
So, we need to find f(0)f(0).

STEP 6

Let's **substitute** x=0x = 0 into our function: f(0)=(0)4+2(0)313(0)210(0)+40f(0) = (0)^4 + 2 \cdot (0)^3 - 13 \cdot (0)^2 - 10 \cdot (0) + 40

STEP 7

Now, let's **simplify**: f(0)=0+000+40f(0) = 0 + 0 - 0 - 0 + 40 f(0)=40f(0) = 40

STEP 8

So, our **yy-intercept** is 40.
As an **ordered pair**, this is (0,40)(0, 40).
Remember, the xx-coordinate is 0 at the yy-intercept!

STEP 9

The **degree** of the polynomial f(x)f(x) is **4**, and the **yy-intercept** is (0,40)(0, 40).

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