Sure, let's solve the division of the polynomial [tex]\(6x^4 - 28x^2 - 9x + 10\)[/tex] by the polynomial [tex]\(x + 2\)[/tex] using long division step by step.
1. Setup the division:
- Dividend: [tex]\(6x^4 - 28x^2 - 9x + 10\)[/tex]
- Divisor: [tex]\(x + 2\)[/tex]
2. Determine the first term of the quotient:
- Divide the leading term of the dividend [tex]\(6x^4\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[
\frac{6x^4}{x} = 6x^3
\][/tex]
- Multiply the entire divisor by this term:
[tex]\[
6x^3 \cdot (x + 2) = 6x^4 + 12x^3
\][/tex]
- Subtract this from the original dividend:
[tex]\[
(6x^4 - 28x^2 - 9x + 10) - (6x^4 + 12x^3) = -12x^3 - 28x^2 - 9x + 10
\][/tex]
3. Determine the second term of the quotient:
- Divide the leading term of the new polynomial [tex]\(-12x^3\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[
\frac{-12x^3}{x} = -12x^2
\][/tex]
- Multiply the entire divisor by this term:
[tex]\[
-12x^2 \cdot (x + 2) = -12x^3 - 24x^2
\][/tex]
- Subtract this from the new polynomial:
[tex]\[
(-12x^3 - 28x^2 - 9x + 10) - (-12x^3 - 24x^2) = -4x^2 - 9x + 10
\][/tex]
4. Determine the third term of the quotient:
- Divide the leading term of the new polynomial [tex]\(-4x^2\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[
\frac{-4x^2}{x} = -4x
\][/tex]
- Multiply the entire divisor by this term:
[tex]\[
-4x \cdot (x + 2) = -4x^2 - 8x
\][/tex]
- Subtract this from the new polynomial:
[tex]\[
(-4x^2 - 9x + 10) - (-4x^2 - 8x) = -x + 10
\][/tex]
5. Determine the fourth term of the quotient:
- Divide the leading term of the new polynomial [tex]\(-x\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[
\frac{-x}{x} = -1
\][/tex]
- Multiply the entire divisor by this term:
[tex]\[
-1 \cdot (x + 2) = -x - 2
\][/tex]
- Subtract this from the new polynomial:
[tex]\[
(-x + 10) - (-x - 2) = 12
\][/tex]
6. Complete the quotient and remainder:
The quotient is the sum of each term derived: [tex]\(6x^3 - 12x^2 - 4x - 1\)[/tex]
The remainder is the result of the final subtraction: [tex]\(12\)[/tex]
Therefore, the quotient is [tex]\(6x^3 - 12x^2 - 4x - 1\)[/tex] and the remainder is [tex]\(12\)[/tex].
So, the division of [tex]\((6x^4 - 28x^2 - 9x + 10) \div (x + 2)\)[/tex] results in:
[tex]\[
6x^3 - 12x^2 - 4x - 1 \quad \text{remainder} \; 12
\][/tex]
