Answer :
To fill in the blanks for the long division process, we need to focus on dividing the polynomial [tex]\(25x^3 + 5x^2 + 14x - 6\)[/tex] by [tex]\(5x - 2\)[/tex]. Let's go through the division step by step:
1. Identify the leading term in the dividend and the divisor:
- For the dividend [tex]\(25x^3 + 5x^2 + 14x - 6\)[/tex], the leading term is [tex]\(25x^3\)[/tex].
- For the divisor [tex]\(5x - 2\)[/tex], the leading term is [tex]\(5x\)[/tex].
2. Divide the leading term of the dividend by the leading term of the divisor:
- Divide [tex]\(25x^3\)[/tex] by [tex]\(5x\)[/tex], which gives [tex]\(5x^2\)[/tex].
3. Write this result above the appropriate term in the dividend:
- You write [tex]\(5x^2\)[/tex] above [tex]\(25x^3\)[/tex] in the dividend.
Therefore, the filled blanks should be:
- Divide [tex]\(25x^3\)[/tex] by [tex]\(5x\)[/tex], which obtains [tex]\(5x^2\)[/tex]. Write this result above [tex]\(25x^3\)[/tex] in the dividend.
If the division process continued, you would follow similar steps, working term by term, reducing the degree of the dividend until all terms have been accounted for.
1. Identify the leading term in the dividend and the divisor:
- For the dividend [tex]\(25x^3 + 5x^2 + 14x - 6\)[/tex], the leading term is [tex]\(25x^3\)[/tex].
- For the divisor [tex]\(5x - 2\)[/tex], the leading term is [tex]\(5x\)[/tex].
2. Divide the leading term of the dividend by the leading term of the divisor:
- Divide [tex]\(25x^3\)[/tex] by [tex]\(5x\)[/tex], which gives [tex]\(5x^2\)[/tex].
3. Write this result above the appropriate term in the dividend:
- You write [tex]\(5x^2\)[/tex] above [tex]\(25x^3\)[/tex] in the dividend.
Therefore, the filled blanks should be:
- Divide [tex]\(25x^3\)[/tex] by [tex]\(5x\)[/tex], which obtains [tex]\(5x^2\)[/tex]. Write this result above [tex]\(25x^3\)[/tex] in the dividend.
If the division process continued, you would follow similar steps, working term by term, reducing the degree of the dividend until all terms have been accounted for.
