6 Questions
What is the formula for finding the sum of zeroes of the quadratic polynomial p(x) = ax^2 + bx + c?
-b/a
According to the HCF and LCM relationship, if HCF(a,b) = 2 and LCM(a,b) = 30, what is a × b?
60
Which of the following is the correct expansion for (a + b + c)^2?
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?
-b/a
What is the simplified form of (a^3 - b^3)?
(a - b)(a^2 + b^2 + ab)
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?
3abc
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
Test your understanding of real numbers, polynomials, quadratic and cubic polynomials, and algebraic identities with this quiz. Covers formulas and concepts like HCF, LCM, sum and product of zeroes, and more.
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