Samacheer Kalvi Class 9 Maths Solutions Chapter 3 Algebra Exercise 3.7

Question 1. Find the quotient and remainder of the following.
(i) \( (4x^3 + 6x^2 - 23x + 18) \div (x + 3) \)
Answer: We use long division to divide the polynomial \( 4x^3 + 6x^2 - 23x + 18 \) by \( x + 3 \). We arrange the terms in descending order of powers of x and then perform the division step-by-step.

The quotient is \( 4x^2 - 6x - 5 \), and the remainder is \( 33 \). This process is useful for factoring polynomials or finding roots when the remainder is zero.
In simple words: When we divide \( 4x^3 + 6x^2 - 23x + 18 \) by \( x + 3 \), we get \( 4x^2 - 6x - 5 \) as the answer with \( 33 \) left over.

🎯 Exam Tip: Always write the terms of the polynomial in descending powers of the variable before starting long division, and be careful with signs when subtracting.

(ii) \( (8y^3 - 16y^2 + 16y - 15) \div (2y - 1) \)
Answer: We perform long division to divide \( 8y^3 - 16y^2 + 16y - 15 \) by \( 2y - 1 \).

The quotient is \( 4y^2 - 6y + 5 \), and the remainder is \( -10 \). This method shows how to break down complex polynomial divisions into simpler steps.
In simple words: Dividing \( 8y^3 - 16y^2 + 16y - 15 \) by \( 2y - 1 \) gives us an answer of \( 4y^2 - 6y + 5 \) with \( -10 \) left over.

🎯 Exam Tip: Pay close attention to the leading coefficients and terms when setting up each step of the long division to avoid errors.

(iii) \( (8x^3 - 1) \div (2x - 1) \)
Answer: To divide \( 8x^3 - 1 \) by \( 2x - 1 \), we first write the polynomial \( 8x^3 - 1 \) as \( 8x^3 + 0x^2 + 0x - 1 \) to include all powers of x. Then we perform long division.

The quotient is \( 4x^2 + 2x + 1 \), and the remainder is \( 0 \). This means \( (2x-1) \) is a factor of \( (8x^3-1) \).
In simple words: When \( 8x^3 - 1 \) is divided by \( 2x - 1 \), the answer is \( 4x^2 + 2x + 1 \), and there is nothing left over.

🎯 Exam Tip: When a polynomial has missing terms, remember to include them with a zero coefficient (like \( 0x^2 \)) to keep the long division aligned correctly.

(iv) \( (-18z + 14z^2 + 24z^3 + 18) \div (3z + 4) \)
Answer: First, we rewrite the dividend \( -18z + 14z^2 + 24z^3 + 18 \) in descending powers of z as \( 24z^3 + 14z^2 - 18z + 18 \). Then we divide by \( 3z + 4 \) using long division.

The quotient is \( 8z^2 - 6z + 2 \), and the remainder is \( 10 \). Always reordering the terms before starting makes the division easier.
In simple words: After arranging the terms, dividing \( 24z^3 + 14z^2 - 18z + 18 \) by \( 3z + 4 \) gives \( 8z^2 - 6z + 2 \) as the answer with \( 10 \) as the leftover.

🎯 Exam Tip: It's crucial to rearrange the polynomial terms from highest to lowest power before starting any division, whether long or synthetic.

 

Question 2. The area of a rectangle is \( x^2 + 7x + 12 \). If its breadth is \( (x + 3) \) then find its length.
Answer: We know that the area of a rectangle is given by the formula: Area = length \( \times \) breadth. Given: Area \( = x^2 + 7x + 12 \) Breadth \( = x + 3 \) To find the length, we divide the area by the breadth: Length \( = \frac{\text{Area}}{\text{Breadth}} \) Length \( = \frac{x^2 + 7x + 12}{x + 3} \) We can factor the numerator or use polynomial long division. Factoring the numerator: \( x^2 + 7x + 12 = (x + 4)(x + 3) \) So, Length \( = \frac{(x + 4)(x + 3)}{x + 3} \) Length \( = x + 4 \) Alternatively, using long division:

The length of the rectangle is \( x + 4 \). This shows how algebraic division can be applied to geometry problems.
In simple words: To find the length, we divide the rectangle's area \( (x^2 + 7x + 12) \) by its breadth \( (x + 3) \). The length of the rectangle is \( x + 4 \).

🎯 Exam Tip: Remember the area formula for a rectangle (length \( \times \) breadth) and use polynomial division or factorization to find a missing side when given the area and one side.

 

Question 3. The base of a parallelogram is \( (5x + 4) \). Find its height if the area is \( 25x^2 - 16 \).
Answer: We know that the area of a parallelogram is given by the formula: Area = base \( \times \) height. Given: Base \( = 5x + 4 \) Area \( = 25x^2 - 16 \) Let the height of the parallelogram be 'h'. So, \( (5x + 4) \times h = 25x^2 - 16 \) To find the height, we divide the area by the base: \( h = \frac{25x^2 - 16}{5x + 4} \) We can recognize the numerator as a difference of squares: \( a^2 - b^2 = (a - b)(a + b) \). Here, \( 25x^2 = (5x)^2 \) and \( 16 = 4^2 \). So, \( 25x^2 - 16 = (5x - 4)(5x + 4) \) Now, substitute this back into the height equation: \( h = \frac{(5x - 4)(5x + 4)}{5x + 4} \) \( h = 5x - 4 \) Alternatively, using long division:

The height of the parallelogram is \( 5x - 4 \). Recognizing special algebraic forms like difference of squares can greatly simplify calculations.
In simple words: Since the area of a parallelogram is base times height, we divide the area \( (25x^2 - 16) \) by the base \( (5x + 4) \) to find the height. The height is \( 5x - 4 \).

🎯 Exam Tip: Always look for opportunities to use algebraic identities (like difference of squares) as they often simplify division problems more quickly than long division.

 

Question 4. The sum of \( (x + 5) \) observations is \( (x^3 + 125) \). Find the mean of the observations.
Answer: The mean of observations is calculated by dividing the sum of observations by the number of observations. Given: Sum of observations \( = x^3 + 125 \) Number of observations \( = x + 5 \) Mean \( = \frac{\text{Sum of observations}}{\text{Number of observations}} \) Mean \( = \frac{x^3 + 125}{x + 5} \) We can recognize the numerator as a sum of cubes: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \). Here, \( x^3 \) and \( 125 = 5^3 \). So, \( x^3 + 125 = (x + 5)(x^2 - 5x + 25) \) Substitute this back into the mean equation: Mean \( = \frac{(x + 5)(x^2 - 5x + 25)}{x + 5} \) Mean \( = x^2 - 5x + 25 \) Alternatively, using long division:

The mean of the observations is \( x^2 - 5x + 25 \). This is a good example of how factoring can simplify polynomial division in statistics.
In simple words: To find the average (mean), we divide the total sum of things \( (x^3 + 125) \) by how many things there are \( (x + 5) \). The average is \( x^2 - 5x + 25 \).

🎯 Exam Tip: When dealing with algebraic expressions for sums and counts, remember the formula for mean and look for algebraic identities like sum or difference of cubes to simplify the division.

 

Question 5. Find the quotient and remainder for the following using synthetic division:
(i) \( (x^3 + x^2 - 7x - 3) \div (x - 3) \)
Answer: For synthetic division, the divisor is \( x - 3 \), so we use \( k = 3 \). The coefficients of the polynomial \( p(x) = x^3 + x^2 - 7x - 3 \) are 1, 1, -7, -3.

The last number in the bottom row (12) is the remainder. The other numbers (1, 4, 5) are the coefficients of the quotient, starting from one degree less than the original polynomial. So, the quotient is \( x^2 + 4x + 5 \), and the remainder is \( 12 \). Synthetic division is a quick way to divide polynomials by linear factors.
In simple words: When we divide \( x^3 + x^2 - 7x - 3 \) by \( x - 3 \) using a shortcut method called synthetic division, we get \( x^2 + 4x + 5 \) as the answer with \( 12 \) left over.

🎯 Exam Tip: Remember that for a divisor \( (x - k) \), you use \( k \) in synthetic division. The coefficients in the bottom row (before the remainder) form the quotient polynomial with one degree less than the original polynomial.

(ii) \( (x^3 + 2x^2 - x - 4) \div (x + 2) \)
Answer: For synthetic division, the divisor is \( x + 2 \), which is \( x - (-2) \), so we use \( k = -2 \). The coefficients of the polynomial \( p(x) = x^3 + 2x^2 - x - 4 \) are 1, 2, -1, -4.

The remainder is \( -2 \). The coefficients of the quotient are 1, 0, -1. So, the quotient is \( x^2 + 0x - 1 = x^2 - 1 \), and the remainder is \( -2 \). This method is very efficient for polynomial division.
In simple words: Using synthetic division to divide \( x^3 + 2x^2 - x - 4 \) by \( x + 2 \) gives \( x^2 - 1 \) as the answer and \( -2 \) as the leftover.

🎯 Exam Tip: When a coefficient in the quotient is 0 (like \( 0x \) here), make sure to write it as part of the polynomial or simply omit it if it's not the highest degree term.

(iii) \( (3x^3 - 2x^2 + 7x - 5) \div (x + 3) \)
Answer: For synthetic division, the divisor is \( x + 3 \), which is \( x - (-3) \), so we use \( k = -3 \). The coefficients of the polynomial \( p(x) = 3x^3 - 2x^2 + 7x - 5 \) are 3, -2, 7, -5.

The remainder is \( -125 \). The coefficients of the quotient are 3, -11, 40. So, the quotient is \( 3x^2 - 11x + 40 \), and the remainder is \( -125 \). This polynomial division method is quick for simple linear divisors.
In simple words: When we divide \( 3x^3 - 2x^2 + 7x - 5 \) by \( x + 3 \) using synthetic division, the answer is \( 3x^2 - 11x + 40 \) and the leftover is \( -125 \).

🎯 Exam Tip: Double-check your arithmetic, especially when multiplying and adding with negative numbers in synthetic division, as sign errors are common.

(iv) \( (8x^4 - 2x^2 + 6x + 5) \div (4x + 1) \)
Answer: For synthetic division, the divisor must be in the form \( x - k \). Here, we have \( 4x + 1 \). First, we divide the entire polynomial by 4 to get a leading coefficient of 1 for the divisor: \( \frac{8x^4 - 2x^2 + 6x + 5}{4x + 1} = \frac{1}{4} \left( \frac{8x^4 - 2x^2 + 6x + 5}{x + \frac{1}{4}} \right) \) The dividend polynomial is \( 8x^4 + 0x^3 - 2x^2 + 6x + 5 \). The divisor is \( x + \frac{1}{4} \), so we use \( k = -\frac{1}{4} \). The coefficients are 8, 0, -2, 6, 5.

The remainder from this division is \( \frac{109}{32} \). The coefficients of the quotient are 8, -2, \( -\frac{3}{2} \), \( \frac{45}{8} \). So the quotient is \( 8x^3 - 2x^2 - \frac{3}{2}x + \frac{45}{8} \). Since we divided the original polynomial by 4 at the start, we must now divide the quotient by 4 to get the actual quotient: Actual Quotient \( = \frac{1}{4} (8x^3 - 2x^2 - \frac{3}{2}x + \frac{45}{8}) \) Actual Quotient \( = 2x^3 - \frac{1}{2}x^2 - \frac{3}{8}x + \frac{45}{32} \) The remainder remains the same, \( \frac{109}{32} \). It is important to remember this final adjustment step when the divisor has a leading coefficient other than 1.
In simple words: For \( (8x^4 - 2x^2 + 6x + 5) \div (4x + 1) \), we first make the divisor simpler by dividing by 4. Then we use synthetic division. The final answer (quotient) is \( 2x^3 - \frac{1}{2}x^2 - \frac{3}{8}x + \frac{45}{32} \), and the leftover (remainder) is \( \frac{109}{32} \).

🎯 Exam Tip: When using synthetic division with a divisor like \( (ax + b) \), divide the entire polynomial by 'a' first. After performing the division, remember to divide the resulting quotient (but not the remainder) by 'a' to get the correct final quotient.

 

Question 6. If the quotient obtained on dividing \( (8x^4 - 2x^2 + 6x - 7) \) by \( (2x + 1) \) is \( (4x^3 + px^2 - qx + 3) \), then find p, q and also the remainder.
Answer: We will use synthetic division to divide \( p(x) = 8x^4 + 0x^3 - 2x^2 + 6x - 7 \) by \( 2x + 1 \). First, we normalize the divisor \( 2x + 1 \) by dividing by 2, which means we use \( k = -\frac{1}{2} \). The coefficients of the polynomial are 8, 0, -2, 6, -7.

The remainder is \( -10 \). The coefficients of the quotient (before dividing by 2) are 8, -4, 0, 6. So, the quotient from the synthetic division is \( 8x^3 - 4x^2 + 0x + 6 \). Now, we must divide this quotient by 2 (since we used \( 2x+1 \) as divisor): Actual Quotient \( = \frac{1}{2} (8x^3 - 4x^2 + 0x + 6) \) Actual Quotient \( = 4x^3 - 2x^2 + 0x + 3 \) We are given that the quotient is \( 4x^3 + px^2 - qx + 3 \). Comparing our calculated quotient \( 4x^3 - 2x^2 + 0x + 3 \) with the given quotient: \( 4x^3 + px^2 - qx + 3 = 4x^3 - 2x^2 + 0x + 3 \) By comparing the coefficients: For \( x^2 \) term: \( p = -2 \) For \( x \) term: \( -q = 0 \implies q = 0 \) The remainder is \( -10 \). This problem shows how comparing polynomial forms helps solve for unknown coefficients.
In simple words: We divide the polynomial \( 8x^4 - 2x^2 + 6x - 7 \) by \( 2x + 1 \) using synthetic division. The remainder is \( -10 \). By comparing the calculated quotient \( (4x^3 - 2x^2 + 0x + 3) \) with the given quotient \( (4x^3 + px^2 - qx + 3) \), we find that \( p = -2 \) and \( q = 0 \).

🎯 Exam Tip: When the divisor is \( ax + b \), remember to adjust the quotient by dividing all its coefficients by 'a' after synthetic division, while the remainder remains unchanged.

 

Question 7. If the quotient obtained on dividing \( 3x^3 + 11x^2 + 34x + 106 \) by \( x - 3 \) is \( 3x^2 + ax + b \), then find a, b and also the remainder.
Answer: We will use synthetic division to divide \( p(x) = 3x^3 + 11x^2 + 34x + 106 \) by \( x - 3 \). For the divisor \( x - 3 \), we use \( k = 3 \). The coefficients of the polynomial are 3, 11, 34, 106.

The remainder is \( 388 \). The coefficients of the quotient are 3, 20, 94. So, the quotient is \( 3x^2 + 20x + 94 \). We are given that the quotient is \( 3x^2 + ax + b \). Comparing our calculated quotient \( 3x^2 + 20x + 94 \) with the given quotient: \( 3x^2 + ax + b = 3x^2 + 20x + 94 \) By comparing the coefficients: For \( x \) term: \( a = 20 \) For constant term: \( b = 94 \) The remainder is \( 388 \). This helps practice synthetic division and comparing polynomial coefficients.
In simple words: Using synthetic division to divide \( 3x^3 + 11x^2 + 34x + 106 \) by \( x - 3 \), we get a quotient of \( 3x^2 + 20x + 94 \). By matching this with the given quotient \( 3x^2 + ax + b \), we find that \( a = 20 \) and \( b = 94 \). The leftover (remainder) is \( 388 \).

🎯 Exam Tip: Carefully align the coefficients of the given and calculated quotients to correctly determine the values of unknown variables like 'a' and 'b'.