**Using the Factor Theorem to Factorize p(x)**

To show that (x-√2) is a factor of the polynomial p(x) = 7x^2 - 4√2x - 6, we can use the factor theorem. The factor theorem states that if a polynomial p(x) has a root r, then (x-r) is a factor of p(x).

**Step 1: Substituting √2 into p(x)**

To begin, we substitute √2 into p(x) and check if the result is equal to zero. If it is, then √2 is a root of p(x), and (x-√2) is a factor.

Substituting √2 into p(x), we have:

p(√2) = 7(√2)^2 - 4√2(√2) - 6

Simplifying further:

p(√2) = 7(2) - 4(2) - 6

p(√2) = 14 - 8 - 6

p(√2) = 0

Since p(√2) equals zero, we can conclude that √2 is a root of p(x).

**Step 2: Using the Factor Theorem**

According to the factor theorem, if √2 is a root of p(x), then (x-√2) is a factor of p(x). Therefore, we can divide p(x) by (x-√2) to obtain the other factor.

**Step 3: Dividing p(x) by (x-√2)**

To divide p(x) by (x-√2), we can use long division or synthetic division. Here, we'll use long division to demonstrate the process.

```
____________________
x-√2 | 7x^2 - 4√2x - 6
- (7x^2 - √2x)
____________________
- 4√2x - 6
+ (4√2x - 2√2)
____________________
- 8√2 - 6
```

The remainder -8√2 - 6 indicates that (x-√2) is indeed a factor of p(x).

**Step 4: Factoring p(x)**

Now that we have confirmed that (x-√2) is a factor of p(x), we can write p(x) as a product of the factors (x-√2) and the quotient we obtained from the division:

p(x) = (x-√2)(7x + 4√2 - 2)

Thus, we have successfully factored p(x) using the factor theorem.

x-√2=0x=√2put the value of x in the polynomial 7×2-4√2x-6=014-4√2×√2-6= 0 14-8-6=0 0=0