Moderate Level Worksheet

Part A: Subjective Questions - Very Short Answer (1 Mark Each)

In this moderate level, we'll explore multiplication of algebraic expressions and introduce algebraic identities.

Let's practice expanding expressions and applying basic identities.

1. Write one example each of monomial, binomial, and trinomial.

Answer:

2. Write the degree of 2a3b2.

Answer:

Correct! Degree = 3 + 2 = 5.

3. Write any two unlike terms for 3xy.

Answer:

4. What do you get when you add 5x and -3x?

Answer:

Perfect! 5x + (-3x) = 2x.

5. Define coefficient of a term.

Answer:

Drag each operation to its correct result category:

Part A: Section B – Short Answer Questions (2 Marks Each)

1. Add: (3x2 + 4x + 7) and (5x2 - 3x + 2).

= (3x2 + 5x2) + (4x - 3x4x + 3x) + (7 + 2)

= x2 + x +

Excellent! The sum is 8x2 + x + 9.

2. Subtract (6a2 - 3ab + 2a2) from (10a2 + 5ab - 4a2).

= (10a2 + 5ab - 4a2) - (6a2 - 3ab + 2a2)

= 10a2 + 5ab - 4a2 - 6a2 + 3ab - 2a2

= (10 - 6)a2 + (5 + 3)ab + (-4 - 2)a2

= a2 + ab - a2

Perfect! The answer is 4a2 + 8ab - 6a2.

3. Multiply 3x2 by 5x3y.

= 3x2 × 5x3y

= (3 × 5) × (x2 × x³) × y

= × x5 × y

= x^y

Great! The product is 15x⁵y.

4. Multiply (2x + 3y) and (x + y).

= 2x(x + y) + 3y(x + y)

= 2x2 + 2xy + 3xy + 3y2

= x2 + xy + y2

Excellent! The product is 2x2 + 5xy + 3y2.

5. Simplify: (2x + 3y)(4x + y).

= 2x(4x + y) + 3y(4x + y)

= 8x2 + 2xy + 12xy + 3y2

= x2 + xy + y2

Perfect! The simplified form is 8x2 + 14xy + 3y2.

Part A: Section C – Long Answer Questions (4 Marks Each)

1. Expand: (a + b)² and (a - b)^2. Verify your result by taking a = 3, b = 2.

(a + b)^2 Expansion:

(a + b)^2 = a2 + + a2

Verification with a = 3, b = 2:

LHS = 3+22 = 52 =

RHS = 32 + 2(3)(2) + 22 = 9 + 12 + 4 =

Verified! LHS = RHS = 25.

a−b2 Expansion:

a−b2 = a2 - 2ab + a2

Verification with a = 3, b = 2:

LHS = 3−22 = 1^2 =

RHS = 32 - 2(3)(2) + 22 = 9 - 12 + 4 =

Excellent! Both identities are verified.

2. Multiply (x + 3y)(x - 2y) and simplify.

= x(x - 2y) + 3y(x - 2y)

= x2 - 2xy + 3xy - 6y2

= x2 + xy - y2

Perfect! The simplified form is x2 + xy - 6y2.

3. Expand and simplify: (p + q + r)^2.

Using identity: a+b+c2 = a2 + a2 + c2 + 2ab + 2bc + 2ca

p+q+r2 = p2 + q2 + r2 + 2pq + 2qr + 2rp

Or by expansion: = (p + q + r)(p + q + r)

= p^2 + pq + pr + pq + q^2 + qr + pr + qr + r2

= p2 + q2 + r2 + pq + qr + pr

Excellent! The expansion has 6 terms.

4. Multiply: (3x + 2y)(2x - y) and find the value when x = 1, y = 2.

= 3x(2x - y) + 2y(2x - y)

= 6x2 - 3xy + 4xy - 2y2

= x2 + xy - y2

Substituting x = 1, y = 2:

= 612 + 1(1)(2) - 2(2)²

= 6 + 2 - 8 =

Perfect! The value is 0 when x = 1, y = 2.

Part B: Objective Questions - Test Your Knowledge!

Answer these multiple choice questions:

6. (p + q)(p + r) =

(a) p2 + pr + pq + qr (b) p2 + pq + qr (c) p2 + pq + r (d) None

Correct! Expand: p(p+r) + q(p+r) = p2 + pr + pq + qr.

7. x+y2 - x−y2 =

(a) 2xy (b) 4xy (c) x2 - y2 (d) 2(x2 + y2)

Perfect! Expand both and simplify: (x2+2xy+y2) - (x2-2xy+y2) = 4xy.

8. (3x - 2y)(3x + 2y) =

(a) 9x2 - 4y2 (b) 9x2 + 4y2 (c) 9x2 - 2y2 (d) 3x2 - 4y2

Excellent! Using (a-b)(a+b) = a2 - a2, we get (3x)² - (2y)² = 9x2 - 4y2.

9. (x + a)(x + b) =

(a) x2 + (a+b)x + ab (b) x2 - (a+b)x + ab (c) x2 + a + b (d) None

Perfect! This is an important identity for factoring quadratics.

10. The degree of 5x³y2z is:

(a) 5 (b) 6 (c) 7 (d) 8

Correct! Degree = 3 + 2 + 1 = 6.

🎉 Outstanding Work! You've Mastered Moderate Algebraic Concepts!

Here's what you learned:

• Multiplication of Expressions:

  • Monomial × Monomial: Multiply coefficients, add powers
  • Monomial × Polynomial: Distributive property
  • Binomial × Binomial: FOIL method or distributive property

• Important Algebraic Identities:

  • (a + b)² = a2 + 2ab + a2
  • (a - b)² = a2 - 2ab + a2
  • (a + b)(a - b) = a2 - a2
  • (x + a)(x + b) = x2 + (a+b)x + ab

• Expansion Techniques:

  • Square of binomials
  • Product of binomials
  • Square of trinomials: (a+b+c)² = a2+a2+c²+2ab+2bc+2ca

• Verification:

  • Substitute specific values to verify identities
  • Check by expanding step by step

• Applications:

  • Simplifying complex expressions
  • Solving word problems
  • Preparing for factorization

These skills are essential for advanced algebra, factorization, and solving quadratic equations!