Questions and Answers
If x + y = 12 and xy = 27, what is the value of x3 + y3?
If (3a + 4b) = 16 and ab = 4, what is the value of (9a2 + 16b2)?
If 4x2 + 9y2 + z2 - 6xy - 3yz - 2xz = 0, prove that 2x = 3y = z.
If a + b + c = 15 and a2 + b2 + c2 = 83, what is the value of a3 + b3 + c3 - 3abc?
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Evaluate 933 - 1073.
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If $a + b = 10$ and $ab = 21$, what is the value of $a^3 + b^3$?
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Given $a^2 + b^2 + c^2 = 20$ and $a + b + c = 0$, what is the value of $ab + bc + ca$?
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If $a + b + c = 14$ and $a^2 + b^2 + c^2 = 74$, find the value of $ab + bc + ca$.
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If $4x + 9y + z - 6xy - 3yz - 2xz = 0$, what relationship holds true between $x$, $y$, and $z$?
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If $a ≠ 2b$ and $a^3 + 8b^3 = 18ab - 27$, what is the value of $a + 2b$?
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Study Notes
Applications of Basic Identities
- System of equations such as (3x + 2y = 12) and (xy = 6) can be used to find specific values, e.g., evaluating (9x^2 + 4y^2).
- Given conditions like (x + y = 12) and (xy = 27), (x^3 + y^3) is computable using the identity (x^3 + y^3 = (x + y)(x^2 - xy + y^2)).
Advanced Applications
- To derive results such as (a^3 + 27b^3 + 54ab = 216), use symmetry and constraints such as (a + 3b = 6).
- Prove identities like (2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca = [(a - b)^2 + (b - c)^2 + (c - a)^2]) by expanding and rearranging terms.
Inequalities and Relations
- Solving linear inequalities like (2x - 1 \leq 0) yields boundaries, displaying (x < \frac{1}{2}).
- Interpreting expressions like (36p^2 + 100q^2 + 4r^2 - 60pq - 20qr - 12pr = 0) leads to specific algebraic relationships among variables (p), (q), and (r).
Evaluations and Calculations
- Complex evaluations, such as (933 - 1073), require mastery of basic arithmetic and understanding negative results.
- Evaluating quadratic expressions and their relationships helps simplify problems effectively, indicated by identities.
Finding Specific Values
- Given equations, find expressions like (a^3 + b^3 + c^3 - 3abc) using known relationships among sums and products of roots.
- The relationship derived from sums (a + b + c = k) and their squares can yield results on individual terms or combinations thereof.
Graphical Representations
- Translating inequalities to number lines, e.g., (x \leq 2 \cup x > 5) results in the range of the solution represented as intervals on the number line.
Systems of Equations
- Finding positive solutions satisfying specific equations like (2a + b + 3c = 6) alongside cubic relationships solidifies understanding of polynomial behavior.
Roots of Polynomial
- Understanding and solving for roots via identities related to conditions imposed on polynomial equations provides insight into their behavior and possible simplifications.
Trigonometric Identities
- Proving trigonometric equalities, such as ((\sin^8\theta - \cos^8\theta) = (\sin^2\theta - \cos^2\theta)(1 - 2\sin^2\theta\cos^2\theta)), enhances calculus and function theory skills.
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Description
Explore the cubic identity involving x, y, and z through factorization. This quiz covers the derivation and application of the formula x³ + y³ + z³ - 3xyz. Test your understanding of polynomial identities and their manipulation.
