CBSE Class 8 Mathematics Algebraic Expressions and Identities Assignment Set 10

VIII - Mathematics Assignment No - 01 - Algebraic Expressions and Identities

Question 1. Classify the following as monomials, binomials and trinomials.
(a) \(3x - 7\)
(b) \(\frac{4}{3}x^2 + 3x - 5\)
(c) \(-10x\)
(d) \(\frac{2}{7}x^2 + 3x + 15\)
(e) \(3xyz\)
(f) \(\frac{3}{5}abcd\)
Answer:
(a) Binomial (contains 2 terms)
(b) Trinomial (contains 3 terms)
(c) Monomial (contains 1 term)
(d) Trinomial (contains 3 terms)
(e) Monomial (contains 1 term)
(f) Monomial (contains 1 term)
In simple words: Expressions with 1 term are monomials, 2 terms are binomials, and 3 terms are trinomials.

Exam Tip: Count the number of unlike terms separated by plus or minus signs to determine the type of polynomial.

Question 2. Write the degree of each of the following expressions.
(a) \(10x\)
(b) \(10x^2 + 5x + 6\)
(c) \(3x^4 - 5x^3 + 7\)
(d) \(4x^2y - 3xy^2 + 5y^3 + x^6\)
Answer:
(a) Degree = 1 (Highest power of variable \(x\) is 1)
(b) Degree = 2 (Highest power of variable \(x\) is 2)
(c) Degree = 4 (Highest power of variable \(x\) is 4)
(d) Degree = 6 (Highest sum of powers in a term is 6 from \(x^6\))
In simple words: The degree is the highest power of the variable in the expression. For multi-variable terms, add the powers of variables in each term.

Exam Tip: Always check all terms to find the maximum exponent or sum of exponents.

Question 3. Write the coefficient of \(x^2\) from the following expressions.
(a) \(3x + 5\)
(b) \(-4x^2 + 5x - 6\)
(c) \(x^3 - 3x^2 + 2x - 9\)
(d) \(x^4 + x^3 + x^2 - 3\)
Answer:
(a) Coefficient of \(x^2\) = 0 (since \(x^2\) term is absent)
(b) Coefficient of \(x^2\) = -4
(c) Coefficient of \(x^2\) = -3
(d) Coefficient of \(x^2\) = 1
In simple words: The coefficient is the numerical number multiplied by \(x^2\). If \(x^2\) is not present, its coefficient is 0.

Exam Tip: Include the negative sign if the term is subtracted.

Question 4. Arrange the following polynomials in descending order.
(a) \(5x^2 - 5x^3 + 5x^6 - 5x^5 + 5x^4 + x + 5\)
(b) \(5x^2y - 4xy + 6xy^4\)
Answer:
(a) Descending order of powers of \(x\):
\(5x^6 - 5x^5 + 5x^4 - 5x^3 + 5x^2 + x + 5\)
(b) Descending order of total degree of terms:
\(6xy^4 + 5x^2y - 4xy\)
In simple words: Write terms from the highest power to the lowest power.

Question 5. Arrange the following polynomials in ascending order.
(a) \(6 + x^2 - 5x + 6x^3\)
(b) \(4x^2y + 5xy^3 - 12x^2y^2 + 15\)
Answer:
(a) Ascending order of powers of \(x\):
\(6 - 5x + x^2 + 6x^3\)
(b) Ascending order of total degree of terms:
\(15 + 5xy^3 + 4x^2y - 12x^2y^2\)
In simple words: Write terms from the lowest power to the highest power.

Exam Tip: Constant terms have a degree of 0, so they come first in ascending order of powers.

Question 6. Add \((a + b + c)\), \((2a - b - c)\) and \((-a + b + c)\).
Answer:
\(= (a + b + c) + (2a - b - c) + (-a + b + c)\)
\(= a + 2a - a + b - b + b + c - c + c\)
\(= 2a + b + c\)
In simple words: Group all like terms together and add their coefficients.

Exam Tip: Be careful with signs when opening parentheses.

Question 7. Add the following polynomials.
(a) \(2a + 3b + c + d\), \(-a + b - c - d\) and \(3a - 4b - 2c - 2d\)
(b) \(3m^2 + 4mn - 5n^2\), \(-m^2 + 2n^2 - 6mn\) and \(3mn - n^2\) and \(3m^2 + 4n^2 - mn\)
Answer:
(a) \(= (2a + 3b + c + d) + (-a + b - c - d) + (3a - 4b - 2c - 2d)\)
\(= (2 - 1 + 3)a + (3 + 1 - 4)b + (1 - 1 - 2)c + (1 - 1 - 2)d\)
\(= 4a + 0b - 2c - 2d\)
\(= 4a - 2c - 2d\)

(b) \(= (3m^2 + 4mn - 5n^2) + (-m^2 + 2n^2 - 6mn) + (3mn - n^2) + (3m^2 + 4n^2 - mn)\)
\(= (3 - 1 + 3)m^2 + (4 - 6 + 3 - 1)mn + (-5 + 2 - 1 + 4)n^2\)
\(= 5m^2 + 0mn + 0n^2\)
\(= 5m^2\)
In simple words: Combine like terms by adding their numerical coefficients.

Exam Tip: Terms that sum to zero can be omitted from the final answer.

Question 8. Subtract.
(a) \(-7x^3 + 8x - 9\) from \(2x^3 - 8x^2 + 9x - 10\)
(b) \(-x^4 + 4x^3y - 6x^2y^2 + 4xy^3 - y^4\) from \(-5x^3y + x^2y^2 - 6y^4\)
Answer:
(a) \(= (2x^3 - 8x^2 + 9x - 10) - (-7x^3 + 8x - 9)\)
\(= 2x^3 - 8x^2 + 9x - 10 + 7x^3 - 8x + 9\)
\(= (2 + 7)x^3 - 8x^2 + (9 - 8)x + (-10 + 9)\)
\(= 9x^3 - 8x^2 + x - 1\)

(b) \(= (-5x^3y + x^2y^2 - 6y^4) - (-x^4 + 4x^3y - 6x^2y^2 + 4xy^3 - y^4)\)
\(= -5x^3y + x^2y^2 - 6y^4 + x^4 - 4x^3y + 6x^2y^2 - 4xy^3 + y^4\)
\(= x^4 + (-5 - 4)x^3y + (1 + 6)x^2y^2 - 4xy^3 + (-6 + 1)y^4\)
\(= x^4 - 9x^3y + 7x^2y^2 - 4xy^3 - 5y^4\)
In simple words: Change the sign of every term in the expression being subtracted, then combine like terms.

Exam Tip: The expression after the word "from" must be written first in the subtraction setup.

Question 9. Subtract the sum of \((a + b + c)\) and \((2a - b + 2c)\) from \(3a - b + 3c\).
Answer:
First, find the sum of \((a + b + c)\) and \((2a - b + 2c)\):
Sum \(= (a + b + c) + (2a - b + 2c)\)
\(= (1 + 2)a + (1 - 1)b + (1 + 2)c\)
\(= 3a + 3c\)
Now, subtract this sum from \(3a - b + 3c\):
\(= (3a - b + 3c) - (3a + 3c)\)
\(= 3a - b + 3c - 3a - 3c\)
\(= (3 - 3)a - b + (3 - 3)c\)
\(= -b\)
In simple words: First add the two expressions, then subtract the resulting sum from the target expression.

Exam Tip: Use parentheses carefully when substituting the calculated sum into the subtraction step.

Question 10. Add \(\frac{2}{3}x^3 - \frac{1}{3}x^2 + \frac{5}{6}x + 7\) and \(\frac{4}{3}x^3 - \frac{2}{3}x^2 - \frac{1}{3}x + 1\).
Answer:
\(= \left(\frac{2}{3}x^3 - \frac{1}{3}x^2 + \frac{5}{6}x + 7\right) + \left(\frac{4}{3}x^3 - \frac{2}{3}x^2 - \frac{1}{3}x + 1\right)\)
Group like terms together:
\(= \left(\frac{2}{3} + \frac{4}{3}\right)x^3 + \left(-\frac{1}{3} - \frac{2}{3}\right)x^2 + \left(\frac{5}{6} - \frac{1}{3}\right)x + (7 + 1)\)
Simplify each coefficient:
\(\frac{2}{3} + \frac{4}{3} = \frac{6}{3} = 2\)
\(-\frac{1}{3} - \frac{2}{3} = -\frac{3}{3} = -1\)
\(\frac{5}{6} - \frac{1}{3} = \frac{5}{6} - \frac{2}{6} = \frac{3}{6} = \frac{1}{2}\)
\(7 + 1 = 8\)
Final Expression \(= 2x^3 - x^2 + \frac{1}{2}x + 8\)
In simple words: Combine fractions with common denominators and add the numerical coefficients for each power of \(x\).

Exam Tip: Take LCM of denominators when combining fractional coefficients.