Math / Algebra

Questioner the following polynomial function. $f(x)=x^{4}+4 x^{3}-3 x^{2}-24 x-18$ of 4 : Factor the polynomial completely.

Studdy Solution

STEP 1

What is this asking? We're asked to completely factor a 4th-degree polynomial, which means we want to rewrite it as a product of simpler terms. Watch out! Factoring higher-degree polynomials can be tricky!
We need to be strategic and use all the tools at our disposal.

STEP 2

1. Find a root
2. Divide by the factor
3. Factor the quotient

STEP 3

Let's try some small values of $x$ to see if we can find a root.
If we plug in $x = -1$ , we get $(-1)^4 + 4(-1)^3 - 3(-1)^2 - 24(-1) - 18 = 1 - 4 - 3 + 24 - 18 = 0.$ Wow, we got lucky! $x = -1$ is a root, so $(x+1)$ is a factor.

STEP 4

Now, let's divide the original polynomial $f(x)$ by $(x+1)$ . $\begin{array}{cccccc} x^3 & +3x^2 & -6x & -18 \\ \hline x+1 & x^4 & +4x^3 & -3x^2 & -24x & -18 \\ x^4 & +x^3 \\ \hline 0 & 3x^3 & -3x^2 \\ & 3x^3 & +3x^2 \\ \hline & 0 & -6x^2 & -24x \\ & & -6x^2 & -6x \\ \hline & & 0 & -18x & -18 \\ & & & -18x & -18 \\ \hline & & & 0 & 0 \end{array}$ So, $f(x) = (x+1)(x^3 + 3x^2 - 6x - 18)$ .

STEP 5

Let's try factoring the cubic by grouping.
We can rewrite the cubic as $x^2(x+3) - 6(x+3).$ Notice that we have a common factor of $(x+3)$ .
Factoring this out, we get $(x+3)(x^2 - 6).$

STEP 6

We can rewrite $x^2 - 6$ as a difference of squares: $x^2 - (\sqrt{6})^2 = (x-\sqrt{6})(x+\sqrt{6})$ .

STEP 7

Putting it all together, the completely factored form of the polynomial is $f(x) = (x+1)(x+3)(x-\sqrt{6})(x+\sqrt{6}).$

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Questioner the following polynomial function. $f(x)=x^{4}+4 x^{3}-3 x^{2}-24 x-18$ of 4 : Factor the polynomial completely.

Studdy Solution

STEP 1

What is this asking? We're asked to completely factor a 4th-degree polynomial, which means we want to rewrite it as a product of simpler terms. Watch out! Factoring higher-degree polynomials can be tricky!
We need to be strategic and use all the tools at our disposal.

STEP 2

1. Find a root
2. Divide by the factor
3. Factor the quotient

STEP 3

Let's try some small values of $x$ to see if we can find a root.
If we plug in $x = -1$ , we get $(-1)^4 + 4(-1)^3 - 3(-1)^2 - 24(-1) - 18 = 1 - 4 - 3 + 24 - 18 = 0.$ Wow, we got lucky! $x = -1$ is a root, so $(x+1)$ is a factor.

STEP 4

STEP 5

Let's try factoring the cubic by grouping.
We can rewrite the cubic as $x^2(x+3) - 6(x+3).$ Notice that we have a common factor of $(x+3)$ .
Factoring this out, we get $(x+3)(x^2 - 6).$

STEP 6

We can rewrite $x^2 - 6$ as a difference of squares: $x^2 - (\sqrt{6})^2 = (x-\sqrt{6})(x+\sqrt{6})$ .

STEP 7

Putting it all together, the completely factored form of the polynomial is $f(x) = (x+1)(x+3)(x-\sqrt{6})(x+\sqrt{6}).$

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Questioner the following polynomial function. f(x)=x4+4x3−3x2−24x−18f(x)=x^{4}+4 x^{3}-3 x^{2}-24 x-18f(x)=x4+4x3−3x2−24x−18 of 4 : Factor the polynomial completely.

Studdy Solution

STEP 1

What is this asking? We're asked to completely factor a 4th-degree polynomial, which means we want to rewrite it as a product of simpler terms. Watch out! Factoring higher-degree polynomials can be tricky!We need to be strategic and use all the tools at our disposal.

STEP 2

1. Find a root2. Divide by the factor3. Factor the quotient

STEP 3

STEP 4

STEP 5

Let's try factoring the cubic by grouping.We can rewrite the cubic as x2(x+3)−6(x+3).x^2(x+3) - 6(x+3).x2(x+3)−6(x+3). Notice that we have a common factor of (x+3)(x+3)(x+3).Factoring this out, we get (x+3)(x2−6).(x+3)(x^2 - 6).(x+3)(x2−6).

STEP 6

We can rewrite x2−6x^2 - 6x2−6 as a difference of squares: x2−(6)2=(x−6)(x+6)x^2 - (\sqrt{6})^2 = (x-\sqrt{6})(x+\sqrt{6})x2−(6​)2=(x−6​)(x+6​).

STEP 7

Putting it all together, the completely factored form of the polynomial is f(x)=(x+1)(x+3)(x−6)(x+6).f(x) = (x+1)(x+3)(x-\sqrt{6})(x+\sqrt{6}).f(x)=(x+1)(x+3)(x−6​)(x+6​).

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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.

Questioner the following polynomial function. $f(x)=x^{4}+4 x^{3}-3 x^{2}-24 x-18$ of 4 : Factor the polynomial completely.

What is this asking? We're asked to completely factor a 4th-degree polynomial, which means we want to rewrite it as a product of simpler terms. Watch out! Factoring higher-degree polynomials can be tricky!
We need to be strategic and use all the tools at our disposal.

1. Find a root
2. Divide by the factor
3. Factor the quotient

Let's try factoring the cubic by grouping.
We can rewrite the cubic as $x^2(x+3) - 6(x+3).$ Notice that we have a common factor of $(x+3)$ .
Factoring this out, we get $(x+3)(x^2 - 6).$

We can rewrite $x^2 - 6$ as a difference of squares: $x^2 - (\sqrt{6})^2 = (x-\sqrt{6})(x+\sqrt{6})$ .

Putting it all together, the completely factored form of the polynomial is $f(x) = (x+1)(x+3)(x-\sqrt{6})(x+\sqrt{6}).$