College

We appreciate your visit to Factor tex f x 3x 3 11x 2 139x 45 tex into linear factors given that 9 is a zero of tex f x tex. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

Factor [tex]f(x)=3x^3+11x^2-139x+45[/tex] into linear factors given that -9 is a zero of [tex]f(x)[/tex].

[tex]f(x)=3x^3+11x^2-139x+45=[/tex]

[tex]\square[/tex]

(Factor completely.)

Answer :

To factor the polynomial [tex]\( f(x) = 3x^3 + 11x^2 - 139x + 45 \)[/tex] into linear factors, given that [tex]\(-9\)[/tex] is a zero of [tex]\( f(x) \)[/tex], follow these steps:

1. Synthetic Division:

Since [tex]\(-9\)[/tex] is a zero of [tex]\( f(x) \)[/tex], [tex]\((x + 9)\)[/tex] is a factor. We will use synthetic division to divide [tex]\( f(x) \)[/tex] by [tex]\((x + 9)\)[/tex].

The coefficients of [tex]\( f(x) \)[/tex] are: [tex]\( 3, 11, -139, 45 \)[/tex].

[tex]\[
\begin{array}{r|rrrr}
-9 & 3 & 11 & -139 & 45 \\
& & -27 & 144 & -45 \\
\hline
& 3 & -16 & 5 & 0 \\
\end{array}
\][/tex]

Synthetic division steps:
- Bring down the first coefficient: [tex]\( 3 \)[/tex].
- Multiply [tex]\(-9\)[/tex] by [tex]\( 3 \)[/tex]: [tex]\( -27 \)[/tex]. Add to the next coefficient: [tex]\( 11 + (-27) = -16 \)[/tex].
- Multiply [tex]\(-9\)[/tex] by [tex]\( -16 \)[/tex]: [tex]\( 144 \)[/tex]. Add to the next coefficient: [tex]\( -139 + 144 = 5 \)[/tex].
- Multiply [tex]\(-9\)[/tex] by [tex]\( 5 \)[/tex]: [tex]\( -45 \)[/tex]. Add to the next coefficient: [tex]\( 45 + (-45) = 0 \)[/tex].

The quotient (result) is [tex]\( 3x^2 - 16x + 5 \)[/tex], and the remainder is [tex]\( 0 \)[/tex].

2. Factoring the Quotient:

Now we need to factor [tex]\( 3x^2 - 16x + 5 \)[/tex].

To factor [tex]\( 3x^2 - 16x + 5 \)[/tex], we look for two numbers that multiply to [tex]\( 3 \cdot 5 = 15 \)[/tex] and add up to [tex]\( -16 \)[/tex].

These numbers are [tex]\( -15 \)[/tex] and [tex]\( -1 \)[/tex].

Rewrite the quadratic expression using these numbers:
[tex]\[
3x^2 - 16x + 5 = 3x^2 - 15x - x + 5
\][/tex]

Factor by grouping:
[tex]\[
3x(x - 5) - 1(x - 5) = (3x - 1)(x - 5)
\][/tex]

3. Compiling the Final Factored Form:

Combining the factor [tex]\( (x + 9) \)[/tex] with the factored form of the quadratic, we get:
[tex]\[
f(x) = (x + 9)(3x - 1)(x - 5)
\][/tex]

So, the completely factored form of the polynomial [tex]\( f(x) = 3x^3 + 11x^2 - 139x + 45 \)[/tex] is:
[tex]\[
f(x) = (x + 9)(3x - 1)(x - 5)
\][/tex]

Thanks for taking the time to read Factor tex f x 3x 3 11x 2 139x 45 tex into linear factors given that 9 is a zero of tex f x tex. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada