Math  /  Algebra

QuestionThe graph of either a cubic or quartic polynomial f(x)f(x) with leading coefficient ±1\pm 1 and integer zeros is shown. Write the complete factored form of f(x)f(x). A. f(x)=(x4)(x+1)(x+3)f(x)=(x-4)(x+1)(x+3) B. f(x)=(x+4)(x1)(x3)f(x)=(x+4)(x-1)(x-3) C. f(x)=(x+4)(x1)(x3)f(x)=-(x+4)(x-1)(x-3) D. f(x)=2(x4)(x+1)(x+3)f(x)=2(x-4)(x+1)(x+3)

Studdy Solution

STEP 1

What is this asking? We need to find the equation of a polynomial, given its graph and knowing the leading coefficient is either 1 or -1. Watch out! The graph can be deceiving!
Don't forget to consider the leading coefficient's sign.

STEP 2

1. Find the roots
2. Build the polynomial
3. Check the leading coefficient

STEP 3

Alright, let's **rock**!
Looking at the graph, we can see where our polynomial **crosses** the x-axis.
These are our **roots**, also known as **zeros** or **x-intercepts**!
We see it crosses at x=4x = -4, x=1x = 1, and x=3x = 3.
Boom!

STEP 4

Now, let's **build** our polynomial.
Since we know the roots, we can write the polynomial in **factored form**.
Each root gives us a factor.
For a root x=ax = a, the corresponding factor is (xa)(x - a).

STEP 5

So, for our roots, we have the factors (x(4))(x - (-4)), (x1)(x - 1), and (x3)(x - 3).
Let's simplify that first factor: (x(4))=(x+4)(x - (-4)) = (x + 4).
This gives us the polynomial f(x)=(x+4)(x1)(x3)f(x) = (x + 4)(x - 1)(x - 3).

STEP 6

We're almost there!
We are told the **leading coefficient** is either 1 or -1.
The leading coefficient is the number multiplying the highest power of xx when the polynomial is expanded.

STEP 7

If we were to **expand** our current polynomial f(x)=(x+4)(x1)(x3)f(x) = (x+4)(x-1)(x-3), the term with the highest power of xx would be xxx=x3x \cdot x \cdot x = x^3, and its coefficient is **1**.
So, our polynomial has a leading coefficient of **1**, which is allowed!

STEP 8

Let's look at the graph again.
As xx goes to positive infinity (way off to the right), the graph also goes to positive infinity (way up).
This confirms that our leading coefficient is indeed positive!

STEP 9

Our final factored form is f(x)=(x+4)(x1)(x3)f(x) = (x + 4)(x - 1)(x - 3), which matches answer choice B!

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