Factoring Cubics using the Factor Theorem (worksheets, videos ... - Free Printable

Let’s solve this step by step.

We are told that 1 is a root of the equation:
> \( x^3 + 6x^2 + 5x - 12 = 0 \)

That means when we plug in \( x = 1 \), the expression equals zero. Let’s verify that first (just to be sure):

\( f(1) = (1)^3 + 6(1)^2 + 5(1) - 12 = 1 + 6 + 5 - 12 = 12 - 12 = 0 \) ✔

So yes, 1 is a root → therefore, \( x - 1 \) is a factor of the polynomial.

Now, we want to divide \( x^3 + 6x^2 + 5x - 12 \) by \( x - 1 \) using polynomial long division — which is what the image shows.

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Step-by-step Long Division:

We divide:
Dividend: \( x^3 + 6x^2 + 5x - 12 \)
Divisor: \( x - 1 \)

#### Step 1: Divide leading term
Divide \( x^3 \) by \( x \) → get \( x^2 \)
Write \( x^2 \) on top.

Multiply \( x^2 \cdot (x - 1) = x^3 - x^2 \)
Subtract from dividend:

```
x^3 + 6x^2 + 5x - 12
- (x^3 - x^2)
-----------------
7x^2 + 5x - 12
```

#### Step 2: Bring down next term (already done above)
Now divide \( 7x^2 \) by \( x \) → get \( 7x \)
Write \( +7x \) on top.

Multiply \( 7x \cdot (x - 1) = 7x^2 - 7x \)
Subtract:

```
7x^2 + 5x - 12
- (7x^2 - 7x)
---------------
12x - 12
```

#### Step 3: Divide \( 12x \) by \( x \) → get \( 12 \)
Write \( +12 \) on top.

Multiply \( 12 \cdot (x - 1) = 12x - 12 \)
Subtract:

```
12x - 12
- (12x - 12)
-------------
0
```

✔ Remainder is 0 — perfect! That confirms \( x - 1 \) is indeed a factor.

The quotient is:
> \( x^2 + 7x + 12 \)

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Now, factor the quadratic if possible:

Factor \( x^2 + 7x + 12 \)

Look for two numbers that multiply to 12 and add to 7 → 3 and 4.

So:
\( x^2 + 7x + 12 = (x + 3)(x + 4) \)

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Therefore, full factorization of original polynomial:

Original: \( x^3 + 6x^2 + 5x - 12 \)
= \( (x - 1)(x^2 + 7x + 12) \)
= \( (x - 1)(x + 3)(x + 4) \)

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Final Answer:

The complete factorization is:
(x - 1)(x + 3)(x + 4)

And since the question was about confirming that \( x - 1 \) is a factor and showing the division — we’ve done that, and found the other factors too.

But based on the context of the image (which stops at the division result), the main output expected is likely the quotient from dividing by \( x - 1 \), which is \( x^2 + 7x + 12 \).

However, since the problem says “e.g. 1 is a root... x-1 is a factor” and then shows the division, and asks us to solve the problem — I believe the goal is to complete the factorization or confirm the process.

Looking again at the user’s request: they uploaded an image with a worked example up to the quotient \( x^2 + 7x + 12 \), and labeled steps 1–4 of long division. The task implied is probably to verify or complete the factorization.

Since no explicit question is written beyond the example, but the title is “FACTOR THEOREM”, and the example ends with the quotient, I think the intended final answer is the fully factored form of the cubic.

Also, note: the handwritten work in the image already got to \( x^2 + 7x + 12 \) as the quotient — so our job is to go one step further and factor that.

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Final Answer:
(x - 1)(x + 3)(x + 4)