Let's work through the polynomial division problems one by one and then verify the relation:
[tex]\[ \text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \][/tex]
### (i) [tex]\(\left(48 x^2+14 x-55\right) \div (8x+9)\)[/tex]
Performing the division:
Quotient: [tex]\(6x - 5\)[/tex]
Remainder: [tex]\(-10\)[/tex]
To verify:
[tex]\[
\text{Dividend} = (8x + 9) \times (6x - 5) + (-10)
\][/tex]
Calculating step-by-step:
1. [tex]\((8x + 9)(6x - 5) = 48x^2 - 40x + 54x - 45 = 48x^2 + 14x - 45\)[/tex]
2. [tex]\(48x^2 + 14x - 45 + (-10) = 48x^2 + 14x - 55\)[/tex]
The calculation verifies the relationship given:
[tex]\[ \text{Dividend was } 48x^2 + 14x - 55 \][/tex]
### (ii) [tex]\(\left(17x^2 + 6x + 2\right) \div (x + 1)\)[/tex]
Performing the division:
Quotient: [tex]\(17x - 11\)[/tex]
Remainder: [tex]\(13\)[/tex]
To verify:
[tex]\[
\text{Dividend} = (x + 1) \times (17x - 11) + 13
\][/tex]
Calculating step-by-step:
1. [tex]\((x + 1)(17x - 11) = 17x^2 - 11x + 17x - 11 = 17x^2 + 6x - 11\)[/tex]
2. [tex]\(17x^2 + 6x - 11 + 13 = 17x^2 + 6x + 2\)[/tex]
The calculation verifies the relationship given:
[tex]\[ \text{Dividend was } 17x^2 + 6x + 2 \][/tex]
### (iii) [tex]\(\left(54a^2 - 15a + 10\right) \div (6a + 5)\)[/tex]
Performing the division:
Quotient: [tex]\(9a - 10\)[/tex]
Remainder: [tex]\(60\)[/tex]
To verify:
[tex]\[
\text{Dividend} = (6a + 5) \times (9a - 10) + 60
\][/tex]
Calculating step-by-step:
1. [tex]\((6a + 5)(9a - 10) = 54a^2 - 60a + 45a - 50 = 54a^2 - 15a - 50\)[/tex]
2. [tex]\(54a^2 - 15a - 50 + 60 = 54a^2 - 15a + 10\)[/tex]
The calculation verifies the relationship given:
[tex]\[ \text{Dividend was } 54a^2 - 15a + 10 \][/tex]
### (iv) [tex]\(\left(3y^2 + 10y + 12\right) \div (y + 3)\)[/tex]
Performing the division:
Quotient: [tex]\(3y + 1\)[/tex]
Remainder: [tex]\(9\)[/tex]
To verify:
[tex]\[
\text{Dividend} = (y + 3) \times (3y + 1) + 9
\][/tex]
Calculating step-by-step:
1. [tex]\((y + 3)(3y + 1) = 3y^2 + y + 9y + 3 = 3y^2 + 10y + 3\)[/tex]
2. [tex]\(3y^2 + 10y + 3 + 9 = 3y^2 + 10y + 12\)[/tex]
The calculation verifies the relationship given:
[tex]\[ \text{Dividend was } 3y^2 + 10y + 12\][/tex]
Thus, in all cases, the relationship:
[tex]\[
\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder}
\][/tex]
is verified successfully.
