Math Snap

STEP 1

What is this asking?
We need to build a polynomial of degree 3, which means it has 3 roots, and we're given the location of the roots.
One root is $3$ and it occurs once, and another root is $1$ and it occurs twice.
Watch out!
Remember that the multiplicity of a root tells us how many times it appears.
Also, the polynomial needs to have integer coefficients and a leading coefficient of 1.

STEP 2

1. Set up the factors
2. Expand the expression
3. Simplify and present

STEP 3

Alright, let's start by writing our polynomial in factored form.
Since the root $3$ has a multiplicity of 1, the factor $(x - 3)$ appears once.
The root $1$ has a multiplicity of 2, so the factor $(x - 1)$ appears twice, which we can write as $(x - 1)^2$.

STEP 4

Putting it all together, our initial polynomial looks like this:
$$f(x) = (x - 3)(x - 1)^2$$ Remember, we're aiming for a polynomial with a leading coefficient of 1, and this form ensures that!

STEP 5

Now, let's expand $(x - 1)^2$ first.
Remember, $(x - 1)^2$ means $(x - 1)(x - 1)$.
Using the FOIL method (First, Outer, Inner, Last), we get:
$$(x - 1)(x - 1) = x \cdot x - x \cdot 1 - 1 \cdot x + 1 \cdot 1 = x^2 - x - x + 1 = x^2 - 2x + 1$$

STEP 6

So, our polynomial becomes:
$$f(x) = (x - 3)(x^2 - 2x + 1)$$

STEP 7

Now, let's expand the entire expression by carefully multiplying each term of $(x - 3)$ with each term of $(x^2 - 2x + 1)$:
\begin{align} f(x) &= x(x^2 - 2x + 1) - 3(x^2 - 2x + 1) \\ &= x^3 - 2x^2 + x - 3x^2 + 6x - 3\end{align}

STEP 8

Finally, let's combine like terms to simplify our polynomial:
$$f(x) = x^3 + (-2x^2 - 3x^2) + (x + 6x) - 3$$ $$f(x) = x^3 - 5x^2 + 7x - 3$$

Our final polynomial is $f(x) = x^3 - 5x^2 + 7x - 3$.
This polynomial has a degree of 3, a leading coefficient of 1, and integer coefficients, just like the problem asked for!