(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a - b)³ = a³ - 3a²b + 3ab² - b³
a³ + b³ = (a + b)(a² - ab + b²)
a³ - b³ = (a - b)(a² + ab + b²)
a² - b² = (a + b)(a - b)
x² - y² = (x + y)(x - y)
(x + y)³ + (x - y)³ = 2x³ + 6xy²
(x + y)³ - (x - y)³ = 6x²y + 2y³
If p(x) is a polynomial and an is a root of p(x), then p(a) = 0
Remainder theorem: If a polynomial p(x) is divided by (x - a), then the remainder is p(a)

🪤 The 5 Mistakes That Cost Marks

Not simplifying the expression before solving
Not using the remainder theorem to find the roots of a polynomial
Not using the factor theorem to factorize a polynomial
Not using the identity (a + b)² = a² + 2ab + b² to simplify expressions
Not checking the degree of the polynomial before solving

✏️ 3 Solved PYQs

Solve: (x + 1)³ - (x - 1)³ Step 1: Use the formula (a + b)³ - (a - b)³ = 6a²b + 2b³ Step 2: Substitute a = x and b = 1 ∈ the formula Step 3: Simplify the expression to get 6x²(1) + 2(1)³ = 6x² + 2
Solve: x³ + 2x² - 7x - 12, given that x + 3 is a factor Step 1: Use the factor theorem to factorize the polynomial Step 2: Divide the polynomial by (x + 3) to get x² - x - 4 Step 3: Factorize the quadratic equation x² - x - 4 to get (x - 2)(x + 2)
Solve: If x + 1 is a factor of x³ + ax² + bx + c, then find the values of a, b, and c Step 1: Use the factor theorem to find the value of c Step 2: Substitute x = -1 ∈ the polynomial to get (-1)³ + a(-1)² + b(-1) + c = 0 Step 3: Simplify the equation to get -1 + a - b + c = 0

🧠 The One Thing Most Students Get Wrong

Most students get the factor theorem wrong, they think that if x - an is a factor of p(x), then p(a) = 1, but actually p(a) = 0

👁️ Ayush's Note

To solve polynomial questions, first try to factorize the polynomial using the factor theorem or the remainder theorem
If the polynomial cannot be factorized, then try to use the formulas for (a + b)², (a - b)², (a + b)³, (a - b)³
Always check the degree of the polynomial before solving

🔁 Last 5 Minutes Box

Make sure to check the degree of the polynomial
Make sure to use the correct formula
Make sure to simplify the expression before solving
Make sure to check the options before marking the answer
Make sure to use the remainder theorem or the factor theorem to find the roots of the polynomial

📝 Practice MCQs

1. What is the value of (x + 1)² - (x - 1)²?

A) 4x

B) 4

C) 4x²

D) 2x + 2

Answer: A) 4x.

2. If x + 2 is a factor of x³ + ax² + bx + c, then what is the value of c?

A) -2

B) -4

C) -6

D) -8

Answer: B) -4.

3. What is the value of x³ + 2x² - 7x - 12, given that x + 3 is a factor?

A) (x + 3)(x² - x - 4)

B) (x + 3)(x² + x - 4)

C) (x - 3)(x² + x - 4)

D) (x - 3)(x² - x - 4)

Answer: A) (x + 3)(x² - x - 4).

4. If x - 2 is a factor of x³ + ax² + bx + c, then what is the value of a?

A) -6

B) -4

C) -2

D) 0

Answer: B) -4.

5. What is the value of (x + 1)³ - (x - 1)³?

A) 6x² + 2

B) 6x² - 2

C) 6x + 2

D) 6x - 2

Answer: A) 6x² + 2.

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This post was curated by Jules, Exam Compass Bot, and edited for accuracy by Ayush.

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(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a - b)³ = a³ - 3a²b + 3ab² - b³
a³ + b³ = (a + b)(a² - ab + b²)
a³ - b³ = (a - b)(a² + ab + b²)
a² - b² = (a + b)(a - b)
x² - y² = (x + y)(x - y)
(x + y)³ + (x - y)³ = 2x³ + 6xy²
(x + y)³ - (x - y)³ = 6x²y + 2y³
If p(x) is a polynomial and an is a root of p(x), then p(a) = 0
Remainder theorem: If a polynomial p(x) is divided by (x - a), then the remainder is p(a)

🪤 The 5 Mistakes That Cost Marks

Not simplifying the expression before solving
Not using the remainder theorem to find the roots of a polynomial
Not using the factor theorem to factorize a polynomial
Not using the identity (a + b)² = a² + 2ab + b² to simplify expressions
Not checking the degree of the polynomial before solving

✏️ 3 Solved PYQs

Solve: (x + 1)³ - (x - 1)³ Step 1: Use the formula (a + b)³ - (a - b)³ = 6a²b + 2b³ Step 2: Substitute a = x and b = 1 ∈ the formula Step 3: Simplify the expression to get 6x²(1) + 2(1)³ = 6x² + 2
Solve: x³ + 2x² - 7x - 12, given that x + 3 is a factor Step 1: Use the factor theorem to factorize the polynomial Step 2: Divide the polynomial by (x + 3) to get x² - x - 4 Step 3: Factorize the quadratic equation x² - x - 4 to get (x - 2)(x + 2)
Solve: If x + 1 is a factor of x³ + ax² + bx + c, then find the values of a, b, and c Step 1: Use the factor theorem to find the value of c Step 2: Substitute x = -1 ∈ the polynomial to get (-1)³ + a(-1)² + b(-1) + c = 0 Step 3: Simplify the equation to get -1 + a - b + c = 0

🧠 The One Thing Most Students Get Wrong

Most students get the factor theorem wrong, they think that if x - an is a factor of p(x), then p(a) = 1, but actually p(a) = 0

👁️ Ayush's Note

To solve polynomial questions, first try to factorize the polynomial using the factor theorem or the remainder theorem
If the polynomial cannot be factorized, then try to use the formulas for (a + b)², (a - b)², (a + b)³, (a - b)³
Always check the degree of the polynomial before solving

🔁 Last 5 Minutes Box

Make sure to check the degree of the polynomial
Make sure to use the correct formula
Make sure to simplify the expression before solving
Make sure to check the options before marking the answer
Make sure to use the remainder theorem or the factor theorem to find the roots of the polynomial

📝 Practice MCQs

1. What is the value of (x + 1)² - (x - 1)²?

A) 4x

B) 4

C) 4x²

D) 2x + 2

Answer: A) 4x.

2. If x + 2 is a factor of x³ + ax² + bx + c, then what is the value of c?

A) -2

B) -4

C) -6

D) -8

Answer: B) -4.

3. What is the value of x³ + 2x² - 7x - 12, given that x + 3 is a factor?

A) (x + 3)(x² - x - 4)

B) (x + 3)(x² + x - 4)

C) (x - 3)(x² + x - 4)

D) (x - 3)(x² - x - 4)

Answer: A) (x + 3)(x² - x - 4).

4. If x - 2 is a factor of x³ + ax² + bx + c, then what is the value of a?

A) -6

B) -4

C) -2

D) 0

Answer: B) -4.

5. What is the value of (x + 1)³ - (x - 1)³?

A) 6x² + 2

B) 6x² - 2

C) 6x + 2

D) 6x - 2

Answer: A) 6x² + 2.

🚀 Ready to Ace Your Exam?

Put your knowledge to the test! Take the free Practice Mock Test now and track your progress against thousands of students.

🎬 Watch video explanations on YouTube →

This post was curated by Jules, Exam Compass Bot, and edited for accuracy by Ayush.

📚 Related Topics

Continue your revision with these related guides: