Answer :
Let's solve the polynomial division of [tex]\((3x^4 - 2x^3 + 7x - 4)\)[/tex] by [tex]\((x - 3)\)[/tex] using synthetic or long division.
### Step-by-Step Long Division:
1. Divide the first term of the dividend [tex]\(3x^4\)[/tex] by the first term of the divisor [tex]\(x\)[/tex]:
[tex]\[
\frac{3x^4}{x} = 3x^3
\][/tex]
Write [tex]\(3x^3\)[/tex] as the first term of the quotient.
2. Multiply the whole divisor [tex]\(x - 3\)[/tex] by this quotient term [tex]\(3x^3\)[/tex]:
[tex]\[
3x^3 \cdot (x - 3) = 3x^4 - 9x^3
\][/tex]
3. Subtract this result from the original dividend:
[tex]\[
(3x^4 - 2x^3 + 7x - 4) - (3x^4 - 9x^3) = 7x^3 + 7x - 4
\][/tex]
4. Repeat the process with the new polynomial [tex]\(7x^3 + 7x - 4\)[/tex]:
Divide the first term [tex]\(7x^3\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{7x^3}{x} = 7x^2
\][/tex]
Write [tex]\(7x^2\)[/tex] as the next term of the quotient.
5. Multiply the divisor by this term [tex]\(7x^2\)[/tex]:
[tex]\[
7x^2 \cdot (x - 3) = 7x^3 - 21x^2
\][/tex]
6. Subtract this from the current polynomial:
[tex]\[
(7x^3 + 7x - 4) - (7x^3 - 21x^2) = 21x^2 + 7x - 4
\][/tex]
7. Continue the process with [tex]\(21x^2 + 7x - 4\)[/tex]:
Divide [tex]\(21x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{21x^2}{x} = 21x
\][/tex]
Write [tex]\(21x\)[/tex] as the next term of the quotient.
8. Multiply the divisor by [tex]\(21x\)[/tex]:
[tex]\[
21x \cdot (x - 3) = 21x^2 - 63x
\][/tex]
9. Subtract:
[tex]\[
(21x^2 + 7x - 4) - (21x^2 - 63x) = 70x - 4
\][/tex]
10. Finally, continue with [tex]\(70x - 4\)[/tex]:
Divide [tex]\(70x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{70x}{x} = 70
\][/tex]
Write [tex]\(70\)[/tex] as the last term of the quotient.
11. Multiply the divisor by [tex]\(70\)[/tex]:
[tex]\[
70 \cdot (x - 3) = 70x - 210
\][/tex]
12. Subtract:
[tex]\[
(70x - 4) - (70x - 210) = 206
\][/tex]
The quotient is [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex] and the remainder is [tex]\(206\)[/tex].
So, the answer is:
a. [tex]\(3x^3 + 7x^2 + 21x + 70: 206\)[/tex]
### Step-by-Step Long Division:
1. Divide the first term of the dividend [tex]\(3x^4\)[/tex] by the first term of the divisor [tex]\(x\)[/tex]:
[tex]\[
\frac{3x^4}{x} = 3x^3
\][/tex]
Write [tex]\(3x^3\)[/tex] as the first term of the quotient.
2. Multiply the whole divisor [tex]\(x - 3\)[/tex] by this quotient term [tex]\(3x^3\)[/tex]:
[tex]\[
3x^3 \cdot (x - 3) = 3x^4 - 9x^3
\][/tex]
3. Subtract this result from the original dividend:
[tex]\[
(3x^4 - 2x^3 + 7x - 4) - (3x^4 - 9x^3) = 7x^3 + 7x - 4
\][/tex]
4. Repeat the process with the new polynomial [tex]\(7x^3 + 7x - 4\)[/tex]:
Divide the first term [tex]\(7x^3\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{7x^3}{x} = 7x^2
\][/tex]
Write [tex]\(7x^2\)[/tex] as the next term of the quotient.
5. Multiply the divisor by this term [tex]\(7x^2\)[/tex]:
[tex]\[
7x^2 \cdot (x - 3) = 7x^3 - 21x^2
\][/tex]
6. Subtract this from the current polynomial:
[tex]\[
(7x^3 + 7x - 4) - (7x^3 - 21x^2) = 21x^2 + 7x - 4
\][/tex]
7. Continue the process with [tex]\(21x^2 + 7x - 4\)[/tex]:
Divide [tex]\(21x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{21x^2}{x} = 21x
\][/tex]
Write [tex]\(21x\)[/tex] as the next term of the quotient.
8. Multiply the divisor by [tex]\(21x\)[/tex]:
[tex]\[
21x \cdot (x - 3) = 21x^2 - 63x
\][/tex]
9. Subtract:
[tex]\[
(21x^2 + 7x - 4) - (21x^2 - 63x) = 70x - 4
\][/tex]
10. Finally, continue with [tex]\(70x - 4\)[/tex]:
Divide [tex]\(70x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{70x}{x} = 70
\][/tex]
Write [tex]\(70\)[/tex] as the last term of the quotient.
11. Multiply the divisor by [tex]\(70\)[/tex]:
[tex]\[
70 \cdot (x - 3) = 70x - 210
\][/tex]
12. Subtract:
[tex]\[
(70x - 4) - (70x - 210) = 206
\][/tex]
The quotient is [tex]\(3x^3 + 7x^2 + 21x + 70\)[/tex] and the remainder is [tex]\(206\)[/tex].
So, the answer is:
a. [tex]\(3x^3 + 7x^2 + 21x + 70: 206\)[/tex]
