Algebra Cubes and Factorization
10 Questions
0 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

If x + y = 12 and xy = 27, what is the value of x3 + y3?

  • 369
  • 288
  • 375 (correct)
  • 384
  • If (3a + 4b) = 16 and ab = 4, what is the value of (9a2 + 16b2)?

  • 50
  • 64
  • 48
  • 56 (correct)
  • If 4x2 + 9y2 + z2 - 6xy - 3yz - 2xz = 0, prove that 2x = 3y = z.

  • False, it proves z = -3y
  • True (correct)
  • False, it proves x = 2y
  • False, it proves 2x = 6y
  • If a + b + c = 15 and a2 + b2 + c2 = 83, what is the value of a3 + b3 + c3 - 3abc?

    <p>410</p> Signup and view all the answers

    Evaluate 933 - 1073.

    <p>156</p> Signup and view all the answers

    If $a + b = 10$ and $ab = 21$, what is the value of $a^3 + b^3$?

    <p>84</p> Signup and view all the answers

    Given $a^2 + b^2 + c^2 = 20$ and $a + b + c = 0$, what is the value of $ab + bc + ca$?

    <p>-20</p> Signup and view all the answers

    If $a + b + c = 14$ and $a^2 + b^2 + c^2 = 74$, find the value of $ab + bc + ca$.

    <p>40</p> Signup and view all the answers

    If $4x + 9y + z - 6xy - 3yz - 2xz = 0$, what relationship holds true between $x$, $y$, and $z$?

    <p>$2x = 3y = z$</p> Signup and view all the answers

    If $a ≠ 2b$ and $a^3 + 8b^3 = 18ab - 27$, what is the value of $a + 2b$?

    <p>2</p> Signup and view all the answers

    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.

    Studying That Suits You

    Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

    Quiz Team

    Related Documents

    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.

    More Like This

    Use Quizgecko on...
    Browser
    Browser