Math / Algebra

QuestionForm a polynomial whose zeros and degree are given. Zeros: 3 , multiplicity 1; 1 , multiplicity 2; degree 3
Type a polynomial with integer coefficients and a leading coefficient of 1 in the box below. $f(x)=$ $\square$ (Simplify your answer.)

Studdy Solution

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: $f(x) = x^3 + (-2x^2 - 3x^2) + (x + 6x) - 3$ $f(x) = x^3 - 5x^2 + 7x - 3$

STEP 9

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!

Was this helpful?

QuestionForm a polynomial whose zeros and degree are given. Zeros: 3 , multiplicity 1; 1 , multiplicity 2; degree 3
Type a polynomial with integer coefficients and a leading coefficient of 1 in the box below. $f(x)=$ $\square$ (Simplify your answer.)

Studdy Solution

STEP 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: $f(x) = x^3 + (-2x^2 - 3x^2) + (x + 6x) - 3$ $f(x) = x^3 - 5x^2 + 7x - 3$

STEP 9

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!

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionForm a polynomial whose zeros and degree are given. Zeros: 3 , multiplicity 1; 1 , multiplicity 2; degree 3Type a polynomial with integer coefficients and a leading coefficient of 1 in the box below. f(x)=f(x)=f(x)= □\square□ (Simplify your answer.)

Studdy Solution

STEP 1

STEP 2

1. Set up the factors2. Expand the expression3. Simplify and present

STEP 3

STEP 4

Putting it all together, our initial polynomial looks like this: f(x)=(x−3)(x−1)2f(x) = (x - 3)(x - 1)^2f(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

STEP 6

So, our polynomial becomes: f(x)=(x−3)(x2−2x+1)f(x) = (x - 3)(x^2 - 2x + 1)f(x)=(x−3)(x2−2x+1)

STEP 7

Now, let's **expand** the entire expression by carefully multiplying each term of (x−3)(x - 3)(x−3) with each term of (x2−2x+1)(x^2 - 2x + 1)(x2−2x+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)=x3+(−2x2−3x2)+(x+6x)−3f(x) = x^3 + (-2x^2 - 3x^2) + (x + 6x) - 3f(x)=x3+(−2x2−3x2)+(x+6x)−3 f(x)=x3−5x2+7x−3f(x) = x^3 - 5x^2 + 7x - 3f(x)=x3−5x2+7x−3

STEP 9

Our final polynomial is f(x)=x3−5x2+7x−3f(x) = x^3 - 5x^2 + 7x - 3f(x)=x3−5x2+7x−3.This polynomial has a degree of **3**, a leading coefficient of **1**, and integer coefficients, just like the problem asked for!

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionForm a polynomial whose zeros and degree are given. Zeros: 3 , multiplicity 1; 1 , multiplicity 2; degree 3
Type a polynomial with integer coefficients and a leading coefficient of 1 in the box below. $f(x)=$ $\square$ (Simplify your answer.)

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

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!

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

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}

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!