Answer :
Sure! Let's break down the polynomial long division step-by-step.
We are given the division problem:
[tex]\[
\frac{25x^3 + 5x^2 + 14x - 6}{5x - 2}
\][/tex]
### Step 1: Divide the leading terms
1. Determine the first term of the quotient:
- Divide the leading term of the dividend [tex]\(25x^3\)[/tex] by the leading term of the divisor [tex]\(5x\)[/tex].
- [tex]\(\frac{25x^3}{5x} = 5x^2\)[/tex].
2. Write [tex]\(5x^2\)[/tex] above the division line over the term [tex]\(25x^3\)[/tex] in the dividend.
### Step 2: Multiply and subtract
1. Multiply the entire divisor [tex]\(5x - 2\)[/tex] by the first term of the quotient [tex]\(5x^2\)[/tex]:
- [tex]\(5x^2 \times (5x - 2) = 25x^3 - 10x^2\)[/tex].
2. Subtract this from the original dividend:
- [tex]\( (25x^3 + 5x^2 + 14x - 6) - (25x^3 - 10x^2) = 15x^2 + 14x - 6\)[/tex].
### Step 3: Repeat the process
1. Next term of the quotient:
- Divide the new leading term [tex]\(15x^2\)[/tex] by the leading term of the divisor [tex]\(5x\)[/tex].
- [tex]\(\frac{15x^2}{5x} = 3x\)[/tex].
2. Write [tex]\(3x\)[/tex] above the division line, next to [tex]\(5x^2\)[/tex].
3. Multiply and subtract:
- Multiply [tex]\(3x \times (5x - 2) = 15x^2 - 6x\)[/tex].
- Subtract: [tex]\((15x^2 + 14x - 6) - (15x^2 - 6x) = 20x - 6\)[/tex].
### Step 4: Repeat again
1. Next term of the quotient:
- Divide the new leading term [tex]\(20x\)[/tex] by the leading term [tex]\(5x\)[/tex].
- [tex]\(\frac{20x}{5x} = 4\)[/tex].
2. Write [tex]\(4\)[/tex] above the division line, next to [tex]\(3x\)[/tex].
3. Multiply and subtract:
- Multiply [tex]\(4 \times (5x - 2) = 20x - 8\)[/tex].
- Subtract: [tex]\((20x - 6) - (20x - 8) = 2\)[/tex].
### Conclusion
- The quotient is [tex]\(5x^2 + 3x + 4\)[/tex].
- The remainder is [tex]\(2\)[/tex].
So, the division can be expressed as:
[tex]\[
\frac{25x^3 + 5x^2 + 14x - 6}{5x - 2} = 5x^2 + 3x + 4 + \frac{2}{5x - 2}
\][/tex]
This is how you complete the polynomial long division for the given expression!
We are given the division problem:
[tex]\[
\frac{25x^3 + 5x^2 + 14x - 6}{5x - 2}
\][/tex]
### Step 1: Divide the leading terms
1. Determine the first term of the quotient:
- Divide the leading term of the dividend [tex]\(25x^3\)[/tex] by the leading term of the divisor [tex]\(5x\)[/tex].
- [tex]\(\frac{25x^3}{5x} = 5x^2\)[/tex].
2. Write [tex]\(5x^2\)[/tex] above the division line over the term [tex]\(25x^3\)[/tex] in the dividend.
### Step 2: Multiply and subtract
1. Multiply the entire divisor [tex]\(5x - 2\)[/tex] by the first term of the quotient [tex]\(5x^2\)[/tex]:
- [tex]\(5x^2 \times (5x - 2) = 25x^3 - 10x^2\)[/tex].
2. Subtract this from the original dividend:
- [tex]\( (25x^3 + 5x^2 + 14x - 6) - (25x^3 - 10x^2) = 15x^2 + 14x - 6\)[/tex].
### Step 3: Repeat the process
1. Next term of the quotient:
- Divide the new leading term [tex]\(15x^2\)[/tex] by the leading term of the divisor [tex]\(5x\)[/tex].
- [tex]\(\frac{15x^2}{5x} = 3x\)[/tex].
2. Write [tex]\(3x\)[/tex] above the division line, next to [tex]\(5x^2\)[/tex].
3. Multiply and subtract:
- Multiply [tex]\(3x \times (5x - 2) = 15x^2 - 6x\)[/tex].
- Subtract: [tex]\((15x^2 + 14x - 6) - (15x^2 - 6x) = 20x - 6\)[/tex].
### Step 4: Repeat again
1. Next term of the quotient:
- Divide the new leading term [tex]\(20x\)[/tex] by the leading term [tex]\(5x\)[/tex].
- [tex]\(\frac{20x}{5x} = 4\)[/tex].
2. Write [tex]\(4\)[/tex] above the division line, next to [tex]\(3x\)[/tex].
3. Multiply and subtract:
- Multiply [tex]\(4 \times (5x - 2) = 20x - 8\)[/tex].
- Subtract: [tex]\((20x - 6) - (20x - 8) = 2\)[/tex].
### Conclusion
- The quotient is [tex]\(5x^2 + 3x + 4\)[/tex].
- The remainder is [tex]\(2\)[/tex].
So, the division can be expressed as:
[tex]\[
\frac{25x^3 + 5x^2 + 14x - 6}{5x - 2} = 5x^2 + 3x + 4 + \frac{2}{5x - 2}
\][/tex]
This is how you complete the polynomial long division for the given expression!
