QuestionQuestion 3
What are the zeros of $f(x)=(2 x+4)(x-1)^{2}$ ? What is its degree and leading coefficient?
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Studdy Solution

STEP 1

What is this asking? We need to find the $\text{x}$ -values that make the function $f(x)$ equal to zero, and also figure out the highest power of $x$ when we expand the function and what number multiplies that highest power of $x$ . Watch out! Don't forget that $(x-1)^2$ means $(x-1)$ multiplied by itself!
Also, remember the degree is the highest power of $x$ after expanding, not before!

STEP 2

1. Find the zeros
2. Find the degree
3. Find the leading coefficient

STEP 3

Alright, so we're looking for the zeros of $f(x) = (2x+4)(x-1)^2$ .
This means we want to find the $x$ values that make $f(x) = 0$ .

STEP 4

Remember, if we have two things multiplied together and they equal zero, then at least one of those things must be zero!
That's our key here.
So, either $2x+4 = 0$ or $(x-1)^2 = 0$ .

STEP 5

Let's tackle $2x+4=0$ first.
We want to isolate $x$ , so let's subtract 4 from both sides: $2x + 4 - 4 = 0 - 4$ , which simplifies to $2x = -4$ .

STEP 6

Now, we divide both sides by 2: $\frac{2x}{2} = \frac{-4}{2}$ .
This gives us $x = -2$ .
Boom! Our first zero is $x = \mathbf{-2}$ .

STEP 7

Next up, $(x-1)^2 = 0$ .
Since squaring something and getting zero means the thing inside must be zero, we have $x-1=0$ .

STEP 8

Add 1 to both sides: $x - 1 + 1 = 0 + 1$ , which gives us $x = \mathbf{1}$ .
Our second zero is $x = \mathbf{1}$ .

STEP 9

To find the degree, we need to expand our function $f(x) = (2x+4)(x-1)^2$ .
First, let's expand $(x-1)^2$ : $(x-1)(x-1) = x^2 -x -x + 1 = x^2 - 2x + 1$ .

STEP 10

Now, multiply this by $(2x+4)$ : $(2x+4)(x^2 - 2x + 1) = 2x(x^2 - 2x + 1) + 4(x^2 - 2x + 1)$ .

STEP 11

Distributing gives us: $2x^3 - 4x^2 + 2x + 4x^2 - 8x + 4$ .
Combining like terms, we get $2x^3 - 6x + 4$ .

STEP 12

The highest power of $x$ is $x^3$ , so the degree of $f(x)$ is $\mathbf{3}$ .

STEP 13

The leading coefficient is the number multiplying the highest power of $x$ .
From our expanded form, $2x^3 - 6x + 4$ , we see that the highest power of $x$ is $x^3$ .

STEP 14

The number multiplying $x^3$ is $\mathbf{2}$ , so our leading coefficient is $\mathbf{2}$ .

STEP 15

The zeros of $f(x)$ are $x = -2$ and $x = 1$ .
The degree of $f(x)$ is 3, and the leading coefficient is 2.

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