Answer :
To factor the polynomial [tex]\( f(x) = 3x^3 + 11x^2 - 139x + 45 \)[/tex] completely given that [tex]\(-9\)[/tex] is a zero, we can follow these steps:
1. Use Polynomial Division:
- Since [tex]\(-9\)[/tex] is a zero, [tex]\( x + 9 \)[/tex] is a factor of the polynomial. We will divide the polynomial by [tex]\( x + 9 \)[/tex].
2. Perform the Division:
- Let's divide [tex]\( 3x^3 + 11x^2 - 139x + 45 \)[/tex] by [tex]\( x + 9 \)[/tex] using synthetic division or long division.
3. Synthetic Division Steps:
- Write down the coefficients of the polynomial: [tex]\( 3, 11, -139, 45 \)[/tex].
- Use [tex]\(-9\)[/tex] as the divisor.
```
-9 | 3 11 -139 45
| -27 144 -45
------------------------
3 -16 5 0
```
- The bottom row gives us the coefficients of the quotient: [tex]\( 3x^2 - 16x + 5 \)[/tex].
- The remainder is [tex]\( 0\)[/tex], confirming that [tex]\( x + 9 \)[/tex] is indeed a factor.
4. Factor the Quotient:
- Now, we need to factor [tex]\( 3x^2 - 16x + 5 \)[/tex].
5. Factoring [tex]\( 3x^2 - 16x + 5 \)[/tex]:
- We will look for two numbers that multiply to [tex]\( 3 \times 5 = 15 \)[/tex] and add up to [tex]\(-16\)[/tex].
- These numbers are [tex]\(-15\)[/tex] and [tex]\(-1\)[/tex].
- Rewrite [tex]\( -16x \)[/tex] as [tex]\( -15x - x \)[/tex]:
[tex]\[
3x^2 - 15x - x + 5
\][/tex]
- Group the terms and factor by grouping:
[tex]\[
(3x^2 - 15x) + (-x + 5)
\][/tex]
- Factor out the common terms:
[tex]\[
3x(x - 5) - 1(x - 5)
\][/tex]
- Factor out the common binomial:
[tex]\[
(x - 5)(3x - 1)
\][/tex]
6. Final Factorization:
- Combine all the factors discovered:
[tex]\[
f(x) = (x + 9)(x - 5)(3x - 1)
\][/tex]
Therefore, the complete factorization of [tex]\( f(x) = 3x^3 + 11x^2 - 139x + 45 \)[/tex] is [tex]\((x + 9)(x - 5)(3x - 1)\)[/tex].
1. Use Polynomial Division:
- Since [tex]\(-9\)[/tex] is a zero, [tex]\( x + 9 \)[/tex] is a factor of the polynomial. We will divide the polynomial by [tex]\( x + 9 \)[/tex].
2. Perform the Division:
- Let's divide [tex]\( 3x^3 + 11x^2 - 139x + 45 \)[/tex] by [tex]\( x + 9 \)[/tex] using synthetic division or long division.
3. Synthetic Division Steps:
- Write down the coefficients of the polynomial: [tex]\( 3, 11, -139, 45 \)[/tex].
- Use [tex]\(-9\)[/tex] as the divisor.
```
-9 | 3 11 -139 45
| -27 144 -45
------------------------
3 -16 5 0
```
- The bottom row gives us the coefficients of the quotient: [tex]\( 3x^2 - 16x + 5 \)[/tex].
- The remainder is [tex]\( 0\)[/tex], confirming that [tex]\( x + 9 \)[/tex] is indeed a factor.
4. Factor the Quotient:
- Now, we need to factor [tex]\( 3x^2 - 16x + 5 \)[/tex].
5. Factoring [tex]\( 3x^2 - 16x + 5 \)[/tex]:
- We will look for two numbers that multiply to [tex]\( 3 \times 5 = 15 \)[/tex] and add up to [tex]\(-16\)[/tex].
- These numbers are [tex]\(-15\)[/tex] and [tex]\(-1\)[/tex].
- Rewrite [tex]\( -16x \)[/tex] as [tex]\( -15x - x \)[/tex]:
[tex]\[
3x^2 - 15x - x + 5
\][/tex]
- Group the terms and factor by grouping:
[tex]\[
(3x^2 - 15x) + (-x + 5)
\][/tex]
- Factor out the common terms:
[tex]\[
3x(x - 5) - 1(x - 5)
\][/tex]
- Factor out the common binomial:
[tex]\[
(x - 5)(3x - 1)
\][/tex]
6. Final Factorization:
- Combine all the factors discovered:
[tex]\[
f(x) = (x + 9)(x - 5)(3x - 1)
\][/tex]
Therefore, the complete factorization of [tex]\( f(x) = 3x^3 + 11x^2 - 139x + 45 \)[/tex] is [tex]\((x + 9)(x - 5)(3x - 1)\)[/tex].
