Question

Factor the polynomial completely: f(x)=x^4+3x^3-3x^2-15x-10

Answer 1

f(x) = (x^2 - 5) (x + 2) (x^2 + 3x - 3)

Explanation:

Okay, factoring the polynomial:

• f(x) = x^4 + 3x^3 - 3x^2 - 15x - 10

⇒ x^4 + 3x^3 - 3x^2 - 15x - 10

⇒ = x^2 (x^2 + 3x - 3) - 5 (x + 2)

⇒ = (x^2 - 5) (x + 2) (x^2 + 3x - 3)

So the fully factored form is:

⇒ f(x) = (x^2 - 5) (x + 2) (x^2 + 3x - 3)

Answer 2

x^2(x - 2)(x + 9) - 12

Explanation:

Sure, here are the steps on how to factor the polynomial completely:

1. **Find the greatest common factor.** The greatest common factor of the polynomial is $x^2$.

2. **Factor out $x^2$.** This gives us the following factorization:

```

f(x) = x^2(x^2 + 3x - 3 - 15)

```

3. **Factor the remaining polynomial.** The remaining polynomial is $x^2 + 3x - 3 - 15 = x^2 + 3x - 18 - 12$. We can factor this polynomial by grouping:

```

x^2 + 3x - 18 - 12 = (x^2 + 3x - 18) + (-12)

= (x - 2)(x + 9) + (-12)

= (x - 2)(x + 9) - 12

```

4. **Putting it all together.** We can now put the entire factorization together:

```

f(x) = x^2(x - 2)(x + 9) - 12

```

Therefore, the complete factorization of the polynomial $f(x) = x^4 + 3x^3 - 3x^2 - 15x - 10$ is $x^2(x - 2)(x + 9) - 12$.

Here is a visual representation of the factorization:

```

f(x) = x^4 + 3x^3 - 3x^2 - 15x - 10 = x^2(x^2 + 3x - 3 - 15) = x^2(x - 2)(x + 9) - 12

```

I hope this helps! Let me know if you have any other questions.

bardAI