Hard Level Worksheet

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

In this hard level, we'll tackle advanced polynomials, complex operations, and challenging algebraic problems.

Let's master the most difficult concepts in algebraic expressions!

1. What is a polynomial?

A polynomial is an algebraic expression with one or moretwo terms

Variables have non-negativeany integerdecimal powers

No variable in the denominatornumerator

Perfect! Polynomials have whole number exponents only.

2. Write the degree of 7x³ + 4x² + 2x + 1.

Highest power of x is 34

Degree = 37

Correct! This is a cubic polynomial (degree 3).

3. Identify the coefficient of x² in 3x² – 4x + 2.

Coefficient of x² = 3–4

Perfect! The coefficient includes the sign.

4. Simplify: 6x – 2x + 3y – y.

Combine x terms: 6x – 2x =

Combine y terms: 3y – y =

Result =

Excellent! Group and simplify like terms.

5. Write the algebraic expression for "sum of squares of x and y".

Square of x = x²2x

Square of y = y²2y

Sum = x² + y²x + y

Perfect! Square each variable, then add.

6. What is the difference between a monomial and binomial?

Monomial has onetwo term(s)

Binomial has twothree term(s)

Examples: (mono), (bi)

Great! The prefixes indicate the number of terms.

7. Find the value of 2x² + 3x + 1 when x = 2.

2()² + 3() + 1

= 2() + + 1

= + 6 + 1

=

Excellent! Follow order of operations.

8. Simplify: (4a + 3b) – (2a – 5b).

Remove brackets: 4a + 3b – 2a + 5b– 5b

Combine a: 4a – 2a =

Combine b: 3b + 5b =

Result =

Perfect! Change signs when subtracting.

9. Write all like terms in: 2x², 3x, 5x², –x.

Like terms with x²: ² and ²

Like terms with x: and

Excellent! Same variable and same power make like terms.

10. Simplify: 3x + 2y + 4x – 3y.

Combine x: 3x + 4x =

Combine y: 2y – 3y =

Result =

Great! 2y – 3y = –1y = –y.

Drag each polynomial to its correct degree category:

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

1. Simplify: (2x² + 3x + 1) + (x² – 2x + 4).

Combine x² terms: 2x² + x² =

Combine x terms: 3x – 2x =

Combine constants: 1 + 4 =

Result =

Perfect! Add coefficients of like terms.

2. Subtract: (4x² – 5x + 3) from (6x² + 2x + 8).

(6x² + 2x + 8) – (4x² – 5x + 3)

= 6x² + 2x + 8 – 4x² + 5x– 5x – 3+ 3

Combine x²: 6x² – 4x² =

Combine x: 2x + 5x =

Combine constants: 8 – 3 =

Result =

Excellent! Change all signs when subtracting.

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

= 3x + 5y + 2 – 2x – 3y+ 3y + 4– 4

Combine x: 3x – 2x =

Combine y: 5y – 3y =

Combine constants: 2 + 4 =

Result =

Great! All signs changed correctly.

4. Add: (2a + 3b + 4) + (4a – b – 5).

Combine a: 2a + 4a =

Combine b: 3b – b =

Combine constants: 4 – 5 =

Result =

Perfect! Remember 3b – b = 2b.

5. Simplify: (4x² + 5x – 2) – (2x² – 3x + 4).

= 4x² + 5x – 2 – 2x² + 3x– 3x – 4+ 4

Combine x²: 4x² – 2x² =

Combine x: 5x + 3x =

Combine constants: –2 – 4 =

Result =

Excellent! Careful with negative signs.

6. Find the value of (2x² + 3x – 5) when x = 3.

Substitute x =

2()² + 3() – 5

= 2() + – 5

= + 9 – 5

=

Perfect! 18 + 9 – 5 = 22.

7. Simplify: (a + 2b – c) + (2a – 3b + c).

Combine a: a + 2a =

Combine b: 2b – 3b =

Combine c: –c + c =

Result = (c terms cancel)

Great! When terms cancel to 0, don't write them.

8. Add: (5x² – 2x + 3) + (x² + 3x – 7).

Combine x²: 5x² + x² =

Combine x: –2x + 3x =

Combine constants: 3 – 7 =

Result =

Perfect! –2x + 3x = 1x = x.

9. Subtract: (3p – 2q + 4) from (5p + q – 2).

(5p + q – 2) – (3p – 2q + 4)

= 5p + q – 2 – 3p + 2q– 2q – 4+ 4

Combine p: 5p – 3p =

Combine q: q + 2q =

Combine constants: –2 – 4 =

Result =

Excellent! All operations performed correctly.

10. Simplify: (6x + 4y – 3) – (2x – y + 5).

= 6x + 4y – 3 – 2x + y– y – 5+ 5

Combine x: 6x – 2x =

Combine y: 4y + y =

Combine constants: –3 – 5 =

Result =

Perfect! All short answer problems complete.

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

1. Simplify and find the value for x = 2, y = 3: (3x + 4y – 5) + (2x – y + 3) – (x + 2y – 4).

Step 1: Add first two expressions

(3x + 4y – 5) + (2x – y + 3)

= + + –22

= 5x + 3y – 2

Step 2: Subtract third expression

(5x + 3y – 2) – (x + 2y – 4)

= 5x + 3y – 2 – x – 2y+ 2y + 4– 4

Combine x: 5x – x =

Combine y: 3y – 2y =

Combine constants: –2 + 4 =

Simplified:

Step 3: Substitute x = 2, y = 3

4() + + 2

= + 3 + 2

=

Excellent! Complete multi-step problem solved.

2. Simplify: (2a + 3b + 4c) – (a – b + 2c) + (3a – 2b + c).

Step 1: Handle subtraction first

(2a + 3b + 4c) – (a – b + 2c)

= 2a + 3b + 4c – a + b– b – 2c+ 2c

= + +

Step 2: Add third expression

(a + 4b + 2c) + (3a – 2b + c)

Combine a: a + 3a =

Combine b: 4b – 2b =

Combine c: 2c + c =

Final answer:

Perfect! Three-term operation completed correctly.

3. Simplify: (5x² + 3x + 2) – (3x² + 2x – 4) + (x² – 4x + 5).

Step 1: Subtract second from first

(5x² + 3x + 2) – (3x² + 2x – 4)

= 5x² + 3x + 2 – 3x² – 2x + 4– 4

= + +

Step 2: Add third expression

(2x² + x + 6) + (x² – 4x + 5)

Combine x²: 2x² + x² =

Combine x: x – 4x =

Combine constants: 6 + 5 =

Final answer:

Excellent! Complex polynomial operation mastered.

4. Add: (3p + 4q – 5r), (2p – q + 3r), and (4p + 3q – 2r).

Combine all p terms:

3p + 2p + 4p =

Combine all q terms:

4q – q + 3q =

Combine all r terms:

–5r + 3r – 2r =

Final answer:

Perfect! Three-expression addition with three variables.

5. Simplify and find the value for a = 2, b = 1: (2a² + 3b + 4) + (a² – 2b – 5) – (3a² + b + 2).

Step 1: Add first two

(2a² + 3b + 4) + (a² – 2b – 5)

= + +

Step 2: Subtract third

(3a² + b – 1) – (3a² + b + 2)

= 3a² + b – 1 – 3a² – b+ b – 2+ 2

Combine a²: 3a² – 3a² =

Combine b: b – b =

Combine constants: –1 – 2 =

Simplified: (all variables cancelled!)

Value: (no need to substitute since no variables remain)

Amazing! Sometimes all variable terms cancel out.

Part B: Objective Questions - Test Your Knowledge!

Answer these multiple choice questions:

6. The number of terms in 2x² + 3x – 4 is:

(a) 2 (b) 3 (c) 4 (d) 1

Correct! The three terms are: 2x², 3x, and –4.

7. Coefficient of y in 4x + 5y – 3 is:

(a) 3 (b) 4 (c) 5 (d) y

Perfect! The coefficient of y is 5.

8. Simplify: (x + 3y – 2) + (2x – y + 4).

(a) 3x + 2y + 2 (b) 3x – 2y + 2 (c) 3x + 4y + 2 (d) x + y + 2

Excellent! x + 2x = 3x, 3y – y = 2y, –2 + 4 = 2.

9. If x = 2, value of 2x² + 3x + 1 is:

(a) 11 (b) 12 (c) 15 (d) 14

Perfect! 2(4) + 3(2) + 1 = 8 + 6 + 1 = 15.

10. The constant term in 3x² – 2x + 7 is:

(a) 7 (b) –2 (c) 3 (d) 5

Correct! The constant term is 7 (no variable attached).

🏆 Outstanding Achievement! You've Mastered Advanced Algebraic Expressions!

Here's what you've conquered at the hard level:

• Advanced Polynomial Concepts:

  Classification by Terms:

  • Monomial: 5x, 3a², –7
  • Binomial: 3x + 5, a² – 4
  • Trinomial: x² + 2x + 1
  • Polynomial: Any expression with multiple terms

  Classification by Degree:

  • Degree 0 (Constant): 5, –3
  • Degree 1 (Linear): 2x + 3
  • Degree 2 (Quadratic): x² + 5x + 6
  • Degree 3 (Cubic): 2x³ + 3x² + x – 1
  • Degree 4 (Quartic): x⁴ + 2x² + 1

• Complex Multi-Step Operations:

  Three-expression problems:

  • Work systematically, two at a time
  • Example: A + B – C
    1. First: A + B
    2. Then: (A + B) – C
  • Keep track of signs at each step

  Variable cancellation:

  • When like terms have equal and opposite coefficients
  • Example: (3x + 5) – (3x + 2) = 5 – 2 = 3
  • Result may be just a constant

• Advanced Substitution:

  With multiple variables:

  • Substitute all values carefully
  • Example: 2x + 3y when x = 2, y = 3
    • = 2(2) + 3(3) = 4 + 9 = 13

  With higher powers:

  • Calculate powers first
  • Example: 2x³ + 5x² + 3x when x = 2
    • = 2(8) + 5(4) + 3(2)
    • = 16 + 20 + 6 = 42

• Critical Sign Management:

  Subtraction rules:

  • Change EVERY sign in the expression being subtracted
  • –(a – b + c) = –a + b – c
  • –(2x² – 3x + 4) = –2x² + 3x – 4

  Multiple operations:

  • Handle one operation at a time
  • Use brackets to organize work
  • Double-check sign changes

• Geometric and Real-World Applications:

  • Perimeter calculations: Sum of all sides
  • Area formulas: Product of dimensions
  • Sum of squares: x² + y² (NOT (x + y)²)
  • Difference of squares: x² – y²
  • Cost calculations: Unit price × quantity + fixed cost

• Problem-Solving Strategies:

  For simplification:

  1. Remove all brackets carefully
  2. Change signs if subtracting
  3. Group all like terms together
  4. Add/subtract coefficients
  5. Write in standard form (highest to lowest degree)
  6. Verify by substituting a test value

  For substitution:

  1. Simplify the expression first
  2. Write the expression with substituted values
  3. Evaluate powers first
  4. Then multiplication
  5. Finally addition/subtraction
  6. Show all steps clearly

• Common Advanced Patterns:

  • a + a + a = 3a (not a³)
  • x × x × x = x³ (not 3x)
  • 2x + 3x – 5x = 0 (terms can cancel)
  • –x means –1x (coefficient is –1)
  • x – 2x = –x (not –2x)
  • (x + y)² ≠ x² + y² (this is a common mistake!)

• Verification Techniques:

  • Substitute x = 1 (simplifies calculation)
  • Check if degree is preserved
  • Verify coefficient signs
  • Count terms before and after
  • Use reverse operation to check

• Critical Mistakes to Avoid:

  • Don't add unlike terms: 3x + 2y ≠ 5xy
  • Don't forget to change all signs when subtracting
  • Don't confuse x² and 2x
  • Don't drop negative signs
  • Don't cancel terms with different variables
  • Don't ignore order of operations in substitution
  • Don't assume (x + y)² = x² + y²

• Advanced Tips:

  • Write expressions in standard form (decreasing powers)
  • Use parentheses to avoid sign errors
  • Check each step before proceeding
  • Simplify before substituting when possible
  • Draw a line under each completed step
  • Circle the final answer

• Connection to Higher Mathematics:

  • These skills lead to solving equations
  • Foundation for factorization
  • Essential for coordinate geometry
  • Used in calculus and beyond
  • Real-world applications in physics, economics, engineering

Mastering algebraic expressions is fundamental to all higher mathematics - you're now ready for equations and advanced algebra!

Remember: Practice is key! Work through 30-40 problems to achieve complete mastery.

Key to success: Accuracy with signs, systematic approach, and careful verification!