Given f(x)= 12x^3 +31x^2 -18x -9, answer the following.   Factor f(x), given that 3/4 is zero.

Given that \( \frac{3}{4} \) is a zero of the polynomial \( f(x) = 12x^3 + 31x^2 - 18x - 9 \), we can factor \( f(x) \) by performing synthetic division or polynomial division.

### Step-by-Step Solution:

#### 1. Use synthetic division with \( \frac{3}{4} \) as a zero.
Since synthetic division with fractions can be tricky, it's often easier to multiply the entire polynomial by 4 to avoid fractions. This step is optional but makes calculations simpler.

The given polynomial is:

\[
f(x) = 12x^3 + 31x^2 - 18x - 9
\]

Since \( \frac{3}{4} \) is a zero, we know that \( x - \frac{3}{4} \) is a factor of \( f(x) \). To avoid fractions in the division process, we multiply the polynomial by 4:

\[
g(x) = 48x^3 + 124x^2 - 72x - 36
\]

Now we can divide \( g(x) \) by \( 4x - 3 \), which corresponds to \( x - \frac{3}{4} \) in simplified form.

#### 2. Perform synthetic division of \( g(x) \) by \( 4x - 3 \).

##### Synthetic Division:

\[
\begin{array}{r|rrrr}
  \frac{3}{4} & 12 & 31 & -18 & -9 \\
    &   & 9 & 30 & 9 \\
\hline
  & 12 & 40 & 12 & 0 \\
\end{array}
\]

The quotient is \( 12x^2 + 40x + 12 \), and the remainder is 0, which verifies that \( \frac{3}{4} \) is indeed a zero of the polynomial.

#### 3. Factor the quotient.

Now, factor the quadratic:

\[
12x^2 + 40x + 12
\]

We can factor out the greatest common factor (GCF), which is 4:

\[
4(3x^2 + 10x + 3)
\]

Next, factor \( 3x^2 + 10x + 3 \):

\[
3x^2 + 10x + 3 = (3x + 1)(x + 3)
\]

#### 4. Write the fully factored form of \( f(x) \).

Thus, the complete factorization of \( f(x) \) is:

\[
f(x) = 4(x - \frac{3}{4})(3x + 1)(x + 3)
\]

Or in simplified form:

\[
f(x) = (4x - 3)(3x + 1)(x + 3)
\]

### Final Answer:
The factored form of \( f(x) = 12x^3 + 31x^2 - 18x - 9 \) is:

\[
f(x) = (4x - 3)(3x + 1)(x + 3)
\]

---

Would you like any more details on this process, or have any questions?

Here are some related questions you could explore:

1. How do you find all the zeros of a cubic function given one zero?
2. What is synthetic division, and how is it used in polynomial division?
3. Can every cubic polynomial be factored, or are there exceptions?
4. How does factoring relate to solving polynomial equations?
5. What are some methods to check if a given factor is correct for a polynomial?

**Tip:** Always test your factorization by multiplying the factors back together to ensure you get the original polynomial.

Given f(x)= 12x^3 +31x^2 -18x -9, answer the following. Factor f(x), given that 3/4 is zero.

Learn how to factor the cubic polynomial f(x) = 12x^3 + 31x^2 - 18x - 9 using synthetic division, given that 3/4 is a zero. This problem involves factoring, synthetic division, and applying the Factor Theorem. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation