Solving Polynomial Equations

Front

What is a polynomial equation?

Back

A polynomial equation is an equation that involves a polynomial expression, which is a mathematical expression consisting of variables raised to whole number powers and coefficients.

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What does it mean to solve a polynomial equation by factoring?

Back

Solving a polynomial equation by factoring involves rewriting the equation as a product of its factors and then setting each factor equal to zero to find the solutions.

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What is the zero-product property?

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The zero-product property states that if the product of two or more factors is zero, then at least one of the factors must be zero.

Front

How do you factor a quadratic equation in the form ax^2 + bx + c?

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To factor a quadratic equation, look for two numbers that multiply to ac (the product of a and c) and add to b. Rewrite the equation using these numbers to split the middle term.

Front

What are the roots of the equation 4x^2 + 11x - 3 = 0?

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The roots of the equation 4x^2 + 11x - 3 = 0 are x = 1/4 and x = -3.

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What is the significance of complex roots in polynomial equations?

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Complex roots indicate that the polynomial does not intersect the x-axis at those points, and they occur in conjugate pairs when the coefficients are real.

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How do you solve the equation x^4 - 256 = 0?

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To solve x^4 - 256 = 0, factor it as (x^2 - 16)(x^2 + 16) = 0, then solve each factor to find x = 4, x = -4, x = 4i, and x = -4i.

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What is the factored form of x^3 + 1?

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The factored form of x^3 + 1 is (x + 1)(x^2 - x + 1).

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What are the solutions to the equation (2x - 1)(x + 4) = 0?

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The solutions to the equation (2x - 1)(x + 4) = 0 are x = 1/2 and x = -4.

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What is the process of synthetic division?

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Synthetic division is a simplified method of dividing a polynomial by a linear factor, which involves using the coefficients of the polynomial and the root of the divisor.