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Questions and Answers
What is the formula for finding the sum of zeroes of the quadratic polynomial p(x) = ax^2 + bx + c?
What is the formula for finding the sum of zeroes of the quadratic polynomial p(x) = ax^2 + bx + c?
- c/a
- -c/a
- b/a
- -b/a (correct)
According to the HCF and LCM relationship, if HCF(a,b) = 2 and LCM(a,b) = 30, what is a × b?
According to the HCF and LCM relationship, if HCF(a,b) = 2 and LCM(a,b) = 30, what is a × b?
- 15
- 30
- 60 (correct)
- 45
Which of the following is the correct expansion for (a + b + c)^2?
Which of the following is the correct expansion for (a + b + c)^2?
- a^2 + b^2 + c^2 + 2(ab + bc - ca)
- a^2 + b^2 + c^2 + 2(ab + bc + ca) (correct)
- a^2 + b^2 + c^2 - 2(ab + bc + ca)
- a^2 + b^2 + c^2 + 2(ab - bc + ca)
For the cubic polynomial p(x) = ax^3 + bx^2 + cx + d, which of these expressions represents the sum of the zeroes?
For the cubic polynomial p(x) = ax^3 + bx^2 + cx + d, which of these expressions represents the sum of the zeroes?
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What is the simplified form of (a^3 - b^3)?
What is the simplified form of (a^3 - b^3)?
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If a, b, and c are real numbers such that a+b+c = 0, what is the value of a^3 + b^3 + c^3?
If a, b, and c are real numbers such that a+b+c = 0, what is the value of a^3 + b^3 + c^3?
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Study Notes
Real Numbers
- HCF (a, b) × LCM (a, b) = a × b
Polynomials
Quadratic Polynomials
- For a quadratic polynomial p(x) = ax² + bx + c, where a ≠0:
- Sum of zeroes = α + β = -b/a
- Product of zeroes = αβ = c/a
Cubic Polynomials
- For a cubic polynomial p(x) = ax³ + bx² + cx + d, where a ≠0:
- Sum of zeroes = α + β + γ = -b/a
- Product of zeroes = αβγ = -d/a
- Sum of product of zeroes taken two at a time = αβ + βγ + γα = c/a
Algebraic Identities
- (a + b)² = a² + b² + 2ab
- (a - b)² = a² + b² - 2ab
- a² - b² = (a + b)(a - b)
- (a + b)³ = a³ + b³ + 3a²b + 3ab²
- (a - b)³ = a³ - b³ - 3a²b + 3ab²
- (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
- a³ - b³ = (a - b)(a² + b² + ab)
- a³ + b³ = (a + b)(a² + b² - ab)
- a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)
Special Case
- If a + b + c = 0, then a³ + b³ + c³ = 3abc
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