To divide the polynomial [tex]\(4x^2 + 3x^3 + 10\)[/tex] by [tex]\(x - 2\)[/tex] using long division, follow these steps:
1. Arrange Polynomials:
Arrange the dividend [tex]\(3x^3 + 4x^2 + 0x + 10\)[/tex] and the divisor [tex]\(x - 2\)[/tex].
2. Divide Leading Terms:
Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{3x^3}{x} = 3x^2
\][/tex]
Write [tex]\(3x^2\)[/tex] above the division line.
3. Multiply and Subtract:
Multiply [tex]\(3x^2\)[/tex] by the divisor [tex]\(x - 2\)[/tex]:
[tex]\[
3x^2 \cdot (x - 2) = 3x^3 - 6x^2
\][/tex]
Subtract this product from the original polynomial:
[tex]\[
3x^3 + 4x^2 + 0x + 10 - (3x^3 - 6x^2) = 10x^2 + 0x + 10
\][/tex]
4. Repeat the Process:
Divide the new leading term [tex]\(10x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{10x^2}{x} = 10x
\][/tex]
Write [tex]\(10x\)[/tex] above the division line next to [tex]\(3x^2\)[/tex].
5. Multiply and Subtract:
Multiply [tex]\(10x\)[/tex] by the divisor [tex]\(x - 2\)[/tex]:
[tex]\[
10x \cdot (x - 2) = 10x^2 - 20x
\][/tex]
Subtract this product from the new polynomial:
[tex]\[
10x^2 + 0x + 10 - (10x^2 - 20x) = 20x + 10
\][/tex]
6. Continue the Process:
Divide the new leading term [tex]\(20x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{20x}{x} = 20
\][/tex]
Write [tex]\(20\)[/tex] above the division line next to [tex]\(10x\)[/tex].
7. Multiply and Subtract:
Multiply [tex]\(20\)[/tex] by the divisor [tex]\(x - 2\)[/tex]:
[tex]\[
20 \cdot (x - 2) = 20x - 40
\][/tex]
Subtract this product from the new polynomial:
[tex]\[
20x + 10 - (20x - 40) = 50
\][/tex]
8. Determine Quotient and Remainder:
The quotient of the division [tex]\( \frac{4x^2 + 3x^3 + 10}{x - 2} \)[/tex] is [tex]\(3x^2 + 10x + 4\)[/tex], and the remainder is [tex]\(2\)[/tex].
So, we write:
[tex]\[
\left(4x^2 + 3x^3 + 10 \right) \div (x - 2) = 3x^2 + 10x + 4 + \frac{2}{x - 2}
\][/tex]
Therefore, the Quotient is:
[tex]\[
3x^2 + 10x + 4 + \left(\frac{2}{x - 2}\right)
\][/tex]
This means that the filled square values in the quotient part of the division are:
[tex]\[
3x^2 + 10x + \boxed{4} + \left(\frac{2}{x-2}\right)
\][/tex]
So, the final step-by-step result for the question is: [tex]\( \boxed{4} \)[/tex] and [tex]\( \boxed{2} \)[/tex].
