Answer :
Let's divide each polynomial and check the relation: Dividend [tex]\( = \)[/tex] Divisor [tex]\( \times \)[/tex] Quotient [tex]\( + \)[/tex] Remainder.
### (i) [tex]\( (48x^2 + 14x - 55) \div (8x + 9) \)[/tex]
#### Step 1: Divide the leading term of the dividend by the leading term of the divisor.
[tex]\[ \frac{48x^2}{8x} = 6x \][/tex]
So, the first term of the quotient is [tex]\( 6x \)[/tex].
#### Step 2: Multiply the entire divisor by this term.
[tex]\[ 6x \times (8x + 9) = 48x^2 + 54x \][/tex]
#### Step 3: Subtract this from the original dividend.
[tex]\[ (48x^2 + 14x - 55) - (48x^2 + 54x) = 14x - 54x - 55 = -40x - 55 \][/tex]
#### Step 4: Divide again.
[tex]\[ \frac{-40x}{8x} = -5 \][/tex]
So, the next term of the quotient is [tex]\( -5 \)[/tex].
#### Step 5: Multiply the entire divisor by this term.
[tex]\[ -5 \times (8x + 9) = -40x - 45 \][/tex]
#### Step 6: Subtract this from the current remainder.
[tex]\[ (-40x - 55) - (-40x - 45) = -55 + 45 = -10 \][/tex]
So, the quotient is [tex]\( 6x - 5 \)[/tex], and the remainder is [tex]\( -10 \)[/tex].
Check Relation:
[tex]\[ 48x^2 + 14x - 55 = (8x + 9)(6x - 5) + (-10) \][/tex]
Expanding the product:
[tex]\[ (8x + 9)(6x - 5) = 48x^2 + 54x - 40x - 45 = 48x^2 + 14x - 45 \][/tex]
Adding the remainder:
[tex]\[ 48x^2 + 14x - 45 + (-10) = 48x^2 + 14x - 55 \][/tex]
The relation holds.
### (ii) [tex]\( (17x^2 + 6x + 2) \div (x + 1) \)[/tex]
#### Step 1: Divide the leading term of the dividend by the leading term of the divisor.
[tex]\[ \frac{17x^2}{x} = 17x \][/tex]
So, the first term of the quotient is [tex]\( 17x \)[/tex].
#### Step 2: Multiply the entire divisor by this term.
[tex]\[ 17x \times (x + 1) = 17x^2 + 17x \][/tex]
#### Step 3: Subtract this from the original dividend.
[tex]\[ (17x^2 + 6x + 2) - (17x^2 + 17x) = 6x - 17x + 2 = -11x + 2 \][/tex]
#### Step 4: Divide again.
[tex]\[ \frac{-11x}{x} = -11 \][/tex]
So, the next term of the quotient is [tex]\( -11 \)[/tex].
#### Step 5: Multiply the entire divisor by this term.
[tex]\[ -11 \times (x + 1) = -11x - 11 \][/tex]
#### Step 6: Subtract this from the current remainder.
[tex]\[ (-11x + 2) - (-11x - 11) = 2 + 11 = 13 \][/tex]
So, the quotient is [tex]\( 17x - 11 \)[/tex], and the remainder is [tex]\( 13 \)[/tex].
Check Relation:
[tex]\[ 17x^2 + 6x + 2 = (x + 1)(17x - 11) + 13 \][/tex]
Expanding the product:
[tex]\[ (x + 1)(17x - 11) = 17x^2 + 17x - 11x - 11 = 17x^2 + 6x - 11 \][/tex]
Adding the remainder:
[tex]\[ 17x^2 + 6x - 11 + 13 = 17x^2 + 6x + 2 \][/tex]
The relation holds.
### (iii) [tex]\( (54a^2 - 15a + 10) \div (6a + 5) \)[/tex]
#### Step 1: Divide the leading term of the dividend by the leading term of the divisor.
[tex]\[ \frac{54a^2}{6a} = 9a \][/tex]
So, the first term of the quotient is [tex]\( 9a \)[/tex].
#### Step 2: Multiply the entire divisor by this term.
[tex]\[ 9a \times (6a + 5) = 54a^2 + 45a \][/tex]
#### Step 3: Subtract this from the original dividend.
[tex]\[ (54a^2 - 15a + 10) - (54a^2 + 45a) = -15a - 45a + 10 = -60a + 10 \][/tex]
#### Step 4: Divide again.
[tex]\[ \frac{-60a}{6a} = -10 \][/tex]
So, the next term of the quotient is [tex]\( -10 \)[/tex].
#### Step 5: Multiply the entire divisor by this term.
[tex]\[ -10 \times (6a + 5) = -60a - 50 \][/tex]
#### Step 6: Subtract this from the current remainder.
[tex]\[ (-60a + 10) - (-60a - 50) = 10 + 50 = 60 \][/tex]
So, the quotient is [tex]\( 9a - 10 \)[/tex], and the remainder is [tex]\( 60 \)[/tex].
Check Relation:
[tex]\[ 54a^2 - 15a + 10 = (6a + 5)(9a - 10) + 60 \][/tex]
Expanding the product:
[tex]\[ (6a + 5)(9a - 10) = 54a^2 - 60a + 45a - 50 = 54a^2 - 15a - 50 \][/tex]
Adding the remainder:
[tex]\[ 54a^2 - 15a - 50 + 60 = 54a^2 - 15a + 10 \][/tex]
The relation holds.
### (iv) [tex]\( (3y^2 + 10y + 12) \div (y + 3) \)[/tex]
#### Step 1: Divide the leading term of the dividend by the leading term of the divisor.
[tex]\[ \frac{3y^2}{y} = 3y \][/tex]
So, the first term of the quotient is [tex]\( 3y \)[/tex].
#### Step 2: Multiply the entire divisor by this term.
[tex]\[ 3y \times (y + 3) = 3y^2 + 9y \][/tex]
#### Step 3: Subtract this from the original dividend.
[tex]\[ (3y^2 + 10y + 12) - (3y^2 + 9y) = 10y - 9y + 12 = y + 12 \][/tex]
#### Step 4: Divide again.
[tex]\[ \frac{y}{y} = 1 \][/tex]
So, the next term of the quotient is [tex]\( 1 \)[/tex].
#### Step 5: Multiply the entire divisor by this term.
[tex]\[ 1 \times (y + 3) = y + 3 \][/tex]
#### Step 6: Subtract this from the current remainder.
[tex]\[ (y + 12) - (y + 3) = 12 - 3 = 9 \][/tex]
So, the quotient is [tex]\( 3y + 1 \)[/tex], and the remainder is [tex]\( 9 \)[/tex].
Check Relation:
[tex]\[ 3y^2 + 10y + 12 = (y + 3)(3y + 1) + 9 \][/tex]
Expanding the product:
[tex]\[ (y + 3)(3y + 1) = 3y^2 + 3y + 9y + 3 = 3y^2 + 10y + 3 \][/tex]
Adding the remainder:
[tex]\[ 3y^2 + 10y + 3 + 9 = 3y^2 + 10y + 12 \][/tex]
The relation holds.
### Summary
For each division problem, the relation:
[tex]\[ \text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \][/tex]
is verified correctly.
### (i) [tex]\( (48x^2 + 14x - 55) \div (8x + 9) \)[/tex]
#### Step 1: Divide the leading term of the dividend by the leading term of the divisor.
[tex]\[ \frac{48x^2}{8x} = 6x \][/tex]
So, the first term of the quotient is [tex]\( 6x \)[/tex].
#### Step 2: Multiply the entire divisor by this term.
[tex]\[ 6x \times (8x + 9) = 48x^2 + 54x \][/tex]
#### Step 3: Subtract this from the original dividend.
[tex]\[ (48x^2 + 14x - 55) - (48x^2 + 54x) = 14x - 54x - 55 = -40x - 55 \][/tex]
#### Step 4: Divide again.
[tex]\[ \frac{-40x}{8x} = -5 \][/tex]
So, the next term of the quotient is [tex]\( -5 \)[/tex].
#### Step 5: Multiply the entire divisor by this term.
[tex]\[ -5 \times (8x + 9) = -40x - 45 \][/tex]
#### Step 6: Subtract this from the current remainder.
[tex]\[ (-40x - 55) - (-40x - 45) = -55 + 45 = -10 \][/tex]
So, the quotient is [tex]\( 6x - 5 \)[/tex], and the remainder is [tex]\( -10 \)[/tex].
Check Relation:
[tex]\[ 48x^2 + 14x - 55 = (8x + 9)(6x - 5) + (-10) \][/tex]
Expanding the product:
[tex]\[ (8x + 9)(6x - 5) = 48x^2 + 54x - 40x - 45 = 48x^2 + 14x - 45 \][/tex]
Adding the remainder:
[tex]\[ 48x^2 + 14x - 45 + (-10) = 48x^2 + 14x - 55 \][/tex]
The relation holds.
### (ii) [tex]\( (17x^2 + 6x + 2) \div (x + 1) \)[/tex]
#### Step 1: Divide the leading term of the dividend by the leading term of the divisor.
[tex]\[ \frac{17x^2}{x} = 17x \][/tex]
So, the first term of the quotient is [tex]\( 17x \)[/tex].
#### Step 2: Multiply the entire divisor by this term.
[tex]\[ 17x \times (x + 1) = 17x^2 + 17x \][/tex]
#### Step 3: Subtract this from the original dividend.
[tex]\[ (17x^2 + 6x + 2) - (17x^2 + 17x) = 6x - 17x + 2 = -11x + 2 \][/tex]
#### Step 4: Divide again.
[tex]\[ \frac{-11x}{x} = -11 \][/tex]
So, the next term of the quotient is [tex]\( -11 \)[/tex].
#### Step 5: Multiply the entire divisor by this term.
[tex]\[ -11 \times (x + 1) = -11x - 11 \][/tex]
#### Step 6: Subtract this from the current remainder.
[tex]\[ (-11x + 2) - (-11x - 11) = 2 + 11 = 13 \][/tex]
So, the quotient is [tex]\( 17x - 11 \)[/tex], and the remainder is [tex]\( 13 \)[/tex].
Check Relation:
[tex]\[ 17x^2 + 6x + 2 = (x + 1)(17x - 11) + 13 \][/tex]
Expanding the product:
[tex]\[ (x + 1)(17x - 11) = 17x^2 + 17x - 11x - 11 = 17x^2 + 6x - 11 \][/tex]
Adding the remainder:
[tex]\[ 17x^2 + 6x - 11 + 13 = 17x^2 + 6x + 2 \][/tex]
The relation holds.
### (iii) [tex]\( (54a^2 - 15a + 10) \div (6a + 5) \)[/tex]
#### Step 1: Divide the leading term of the dividend by the leading term of the divisor.
[tex]\[ \frac{54a^2}{6a} = 9a \][/tex]
So, the first term of the quotient is [tex]\( 9a \)[/tex].
#### Step 2: Multiply the entire divisor by this term.
[tex]\[ 9a \times (6a + 5) = 54a^2 + 45a \][/tex]
#### Step 3: Subtract this from the original dividend.
[tex]\[ (54a^2 - 15a + 10) - (54a^2 + 45a) = -15a - 45a + 10 = -60a + 10 \][/tex]
#### Step 4: Divide again.
[tex]\[ \frac{-60a}{6a} = -10 \][/tex]
So, the next term of the quotient is [tex]\( -10 \)[/tex].
#### Step 5: Multiply the entire divisor by this term.
[tex]\[ -10 \times (6a + 5) = -60a - 50 \][/tex]
#### Step 6: Subtract this from the current remainder.
[tex]\[ (-60a + 10) - (-60a - 50) = 10 + 50 = 60 \][/tex]
So, the quotient is [tex]\( 9a - 10 \)[/tex], and the remainder is [tex]\( 60 \)[/tex].
Check Relation:
[tex]\[ 54a^2 - 15a + 10 = (6a + 5)(9a - 10) + 60 \][/tex]
Expanding the product:
[tex]\[ (6a + 5)(9a - 10) = 54a^2 - 60a + 45a - 50 = 54a^2 - 15a - 50 \][/tex]
Adding the remainder:
[tex]\[ 54a^2 - 15a - 50 + 60 = 54a^2 - 15a + 10 \][/tex]
The relation holds.
### (iv) [tex]\( (3y^2 + 10y + 12) \div (y + 3) \)[/tex]
#### Step 1: Divide the leading term of the dividend by the leading term of the divisor.
[tex]\[ \frac{3y^2}{y} = 3y \][/tex]
So, the first term of the quotient is [tex]\( 3y \)[/tex].
#### Step 2: Multiply the entire divisor by this term.
[tex]\[ 3y \times (y + 3) = 3y^2 + 9y \][/tex]
#### Step 3: Subtract this from the original dividend.
[tex]\[ (3y^2 + 10y + 12) - (3y^2 + 9y) = 10y - 9y + 12 = y + 12 \][/tex]
#### Step 4: Divide again.
[tex]\[ \frac{y}{y} = 1 \][/tex]
So, the next term of the quotient is [tex]\( 1 \)[/tex].
#### Step 5: Multiply the entire divisor by this term.
[tex]\[ 1 \times (y + 3) = y + 3 \][/tex]
#### Step 6: Subtract this from the current remainder.
[tex]\[ (y + 12) - (y + 3) = 12 - 3 = 9 \][/tex]
So, the quotient is [tex]\( 3y + 1 \)[/tex], and the remainder is [tex]\( 9 \)[/tex].
Check Relation:
[tex]\[ 3y^2 + 10y + 12 = (y + 3)(3y + 1) + 9 \][/tex]
Expanding the product:
[tex]\[ (y + 3)(3y + 1) = 3y^2 + 3y + 9y + 3 = 3y^2 + 10y + 3 \][/tex]
Adding the remainder:
[tex]\[ 3y^2 + 10y + 3 + 9 = 3y^2 + 10y + 12 \][/tex]
The relation holds.
### Summary
For each division problem, the relation:
[tex]\[ \text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \][/tex]
is verified correctly.
