Answer :
Let's solve the problem of dividing the polynomial [tex]\(10x^3 + 28x^2 + 12x - 37\)[/tex] by [tex]\(2x + 4\)[/tex] using polynomial long division. Here’s a step-by-step explanation:
1. Set up the Division: Write [tex]\(10x^3 + 28x^2 + 12x - 37\)[/tex] as the dividend (under the division symbol) and [tex]\(2x + 4\)[/tex] as the divisor (outside).
2. Divide the First Term:
- Divide the leading term of the dividend, [tex]\(10x^3\)[/tex], by the leading term of the divisor, [tex]\(2x\)[/tex].
- This gives [tex]\(\frac{10x^3}{2x} = 5x^2\)[/tex].
3. Multiply and Subtract:
- Multiply [tex]\(5x^2\)[/tex] by the entire divisor [tex]\(2x + 4\)[/tex], yielding [tex]\(5x^2 \cdot (2x + 4) = 10x^3 + 20x^2\)[/tex].
- Subtract this from the original dividend:
[tex]\[
(10x^3 + 28x^2) - (10x^3 + 20x^2) = 8x^2.
\][/tex]
- Bring down the next term from the dividend, resulting in [tex]\(8x^2 + 12x\)[/tex].
4. Repeat the Steps:
- Divide: Divide the new leading term [tex]\(8x^2\)[/tex] by [tex]\(2x\)[/tex], resulting in [tex]\(\frac{8x^2}{2x} = 4x\)[/tex].
- Multiply and Subtract: Multiply [tex]\(4x\)[/tex] by the divisor:
[tex]\[
4x \cdot (2x + 4) = 8x^2 + 16x.
\][/tex]
Subtract:
[tex]\[
(8x^2 + 12x) - (8x^2 + 16x) = -4x.
\][/tex]
- Bring down the next term, getting [tex]\(-4x - 37\)[/tex].
5. Continue Once More:
- Divide: Divide [tex]\(-4x\)[/tex] by [tex]\(2x\)[/tex] to get [tex]\(-2\)[/tex].
- Multiply and Subtract: Multiply [tex]\(-2\)[/tex] by the divisor:
[tex]\[
-2 \cdot (2x + 4) = -4x - 8.
\][/tex]
Subtract:
[tex]\[
(-4x - 37) - (-4x - 8) = -29.
\][/tex]
6. Write the Result:
- The quotient is [tex]\(5x^2 + 4x - 2\)[/tex].
- The remainder is [tex]\(-29\)[/tex].
Now, we can express the division as:
[tex]\[
10x^3 + 28x^2 + 12x - 37 = (2x + 4)(5x^2 + 4x - 2) - 29.
\][/tex]
This completes the division process, and we have the polynomial equation in the desired form.
1. Set up the Division: Write [tex]\(10x^3 + 28x^2 + 12x - 37\)[/tex] as the dividend (under the division symbol) and [tex]\(2x + 4\)[/tex] as the divisor (outside).
2. Divide the First Term:
- Divide the leading term of the dividend, [tex]\(10x^3\)[/tex], by the leading term of the divisor, [tex]\(2x\)[/tex].
- This gives [tex]\(\frac{10x^3}{2x} = 5x^2\)[/tex].
3. Multiply and Subtract:
- Multiply [tex]\(5x^2\)[/tex] by the entire divisor [tex]\(2x + 4\)[/tex], yielding [tex]\(5x^2 \cdot (2x + 4) = 10x^3 + 20x^2\)[/tex].
- Subtract this from the original dividend:
[tex]\[
(10x^3 + 28x^2) - (10x^3 + 20x^2) = 8x^2.
\][/tex]
- Bring down the next term from the dividend, resulting in [tex]\(8x^2 + 12x\)[/tex].
4. Repeat the Steps:
- Divide: Divide the new leading term [tex]\(8x^2\)[/tex] by [tex]\(2x\)[/tex], resulting in [tex]\(\frac{8x^2}{2x} = 4x\)[/tex].
- Multiply and Subtract: Multiply [tex]\(4x\)[/tex] by the divisor:
[tex]\[
4x \cdot (2x + 4) = 8x^2 + 16x.
\][/tex]
Subtract:
[tex]\[
(8x^2 + 12x) - (8x^2 + 16x) = -4x.
\][/tex]
- Bring down the next term, getting [tex]\(-4x - 37\)[/tex].
5. Continue Once More:
- Divide: Divide [tex]\(-4x\)[/tex] by [tex]\(2x\)[/tex] to get [tex]\(-2\)[/tex].
- Multiply and Subtract: Multiply [tex]\(-2\)[/tex] by the divisor:
[tex]\[
-2 \cdot (2x + 4) = -4x - 8.
\][/tex]
Subtract:
[tex]\[
(-4x - 37) - (-4x - 8) = -29.
\][/tex]
6. Write the Result:
- The quotient is [tex]\(5x^2 + 4x - 2\)[/tex].
- The remainder is [tex]\(-29\)[/tex].
Now, we can express the division as:
[tex]\[
10x^3 + 28x^2 + 12x - 37 = (2x + 4)(5x^2 + 4x - 2) - 29.
\][/tex]
This completes the division process, and we have the polynomial equation in the desired form.
