Math Problem Statement
Solution
The image contains two problems involving cubic equations (third-degree equations) to solve and verify. Let's break them down:
Problem 1:
Given the function , the goal is to solve the equation .
Steps:
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a) Verify that is a solution of the equation .
This means substituting into the equation and checking if the result is 0.
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b) Show that the cubic function can be written in the form by expanding and identifying the coefficients of , , and .
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c) Solve the quadratic equation .
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d) Deduce all the solutions of the equation , by factoring it completely.
Problem 2:
Solve the equation .
Steps:
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a) Verify that is a solution of this equation. Then factor the cubic polynomial by expressing it as , where is a quadratic polynomial.
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b) Find all the solutions of the equation by solving the quadratic polynomial .
Do you want me to walk you through each step in detail, or would you prefer specific assistance with any part of these problems?
Relative Questions:
- How do we confirm whether a given value is a solution to a cubic equation?
- What is the process for factoring a cubic polynomial?
- How can we determine the coefficients when rewriting a cubic equation as a product of a linear and quadratic term?
- What methods can we use to solve quadratic equations?
- How can we use factoring to find all solutions of a cubic equation?
Tip:
When verifying if a specific value is a solution, always substitute the value into the equation and simplify to check if the result is zero.
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Math Problem Analysis
Mathematical Concepts
Cubic Equations
Factoring
Quadratic Equations
Verification of Roots
Formulas
Cubic equation: f(x) = ax^3 + bx^2 + cx + d = 0
Factoring: (x - root)(quadratic polynomial)
Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
Theorems
Rational Root Theorem
Fundamental Theorem of Algebra
Suitable Grade Level
Grades 10-12