Answer :

To divide the polynomial [tex]\(4x^3 - 19x^2 + x - 30\)[/tex] by [tex]\(x - 5\)[/tex], we can use synthetic division. Let's go through the process step by step:

1. Identify the coefficients:
- Write down the coefficients of the polynomial: [tex]\(4, -19, 1, -30\)[/tex].

2. Set up the synthetic division:
- You'll divide by [tex]\(x - 5\)[/tex], so use the number 5 in the synthetic division process.

3. Perform the synthetic division:
- Bring down the first coefficient (4) as it is.
- Multiply this value (4) by 5 (the number we're dividing by) and write the result under the next coefficient.
- Add this value to the next coefficient: [tex]\(-19 + 20 = 1\)[/tex].
- Repeat this process: multiply the new result (1) by 5 and add it to the next coefficient.
- [tex]\(1 \times 5 = 5\)[/tex] and [tex]\(1 + 5 = 6\)[/tex].
- Finally, multiply 6 by 5 to get 30, and add this to the last coefficient: [tex]\(-30 + 30 = 0\)[/tex].

4. Interpret the results:
- The numbers you get at the bottom row are the coefficients of the quotient.
- The final number is the remainder.

So, the quotient from dividing the polynomial is [tex]\(4x^2 + 1x + 6\)[/tex], and the remainder is 0.

Therefore,

[tex]\[
(4x^3 - 19x^2 + x - 30) \div (x - 5) = 4x^2 + x + 6
\][/tex]

This confirms that [tex]\(4x^2 + x + 6\)[/tex] is the result of the division, with a remainder of 0.

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Rewritten by : Barada