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Factor [tex]\( f(x) = 4x^3 + 19x^2 - 101x + 24 \)[/tex] into linear factors given that [tex]\(-8\)[/tex] is a zero of [tex]\( f(x) \)[/tex].

[tex]\[ f(x) = 4x^3 + 19x^2 - 101x + 24 = \boxed{\text{(Factor completely)}} \][/tex]

(Factor completely.)

Answer :

To factor the polynomial [tex]\( f(x) = 4x^3 + 19x^2 - 101x + 24 \)[/tex] completely, given that [tex]\( -8 \)[/tex] is a zero of [tex]\( f(x) \)[/tex], follow these steps:

1. Synthetic Division: Use synthetic division to divide the polynomial [tex]\( f(x) \)[/tex] by [tex]\( x + 8 \)[/tex].

Here's how synthetic division works:

- Write down the coefficients of the polynomial: [tex]\( 4, 19, -101, 24 \)[/tex].
- Write the zero of the polynomial [tex]\( x = -8 \)[/tex] to the left.

Perform the synthetic division steps:
```
-8 | 4 19 -101 24
| -32 104 -24
----------------------
4 -13 3 0
```

- Bring down the first coefficient (4).
- Multiply [tex]\( 4 \)[/tex] by [tex]\( -8 \)[/tex] to get [tex]\( -32 \)[/tex], and add it to the next coefficient [tex]\( 19 \)[/tex] to get [tex]\( -13 \)[/tex].
- Multiply [tex]\( -13 \)[/tex] by [tex]\( -8 \)[/tex] to get [tex]\( 104 \)[/tex], and add it to the next coefficient [tex]\( -101 \)[/tex] to get [tex]\( 3 \)[/tex].
- Multiply [tex]\( 3 \)[/tex] by [tex]\( -8 \)[/tex] to get [tex]\( -24 \)[/tex], and add it to the next coefficient [tex]\( 24 \)[/tex] to get [tex]\( 0 \)[/tex].

The result of the synthetic division is:
[tex]\[
4x^2 - 13x + 3
\][/tex]
And the remainder is [tex]\( 0 \)[/tex], which confirms that [tex]\( -8 \)[/tex] is indeed a zero of [tex]\( f(x) \)[/tex].

2. Factor the resulting quadratic equation: Now factor the quadratic polynomial [tex]\( 4x^2 - 13x + 3 \)[/tex].

We seek two numbers that multiply to [tex]\( 4 \times 3 = 12 \)[/tex] and add up to [tex]\( -13 \)[/tex]. These numbers are [tex]\( -12 \)[/tex] and [tex]\( -1 \)[/tex].

Rewrite the middle term using these numbers:
[tex]\[
4x^2 - 12x - x + 3
\][/tex]

Factor by grouping:
[tex]\[
4x^2 - 12x - x + 3 = 4x(x - 3) - 1(x - 3)
\][/tex]

Factor out the common factor [tex]\((x - 3)\)[/tex]:
[tex]\[
(4x - 1)(x - 3)
\][/tex]

3. Combine the factors: Now, combine the linear factor by [tex]\( x + 8 \)[/tex] with the factors of the quadratic polynomial:

[tex]\[
f(x) = (x + 8)(4x - 1)(x - 3)
\][/tex]

So, the polynomial [tex]\( f(x) = 4x^3 + 19x^2 - 101x + 24 \)[/tex] factors completely as:

[tex]\[
f(x) = (x + 8)(4x - 1)(x - 3)
\][/tex]

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