Solving Polynomial Equations

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What is a polynomial equation?

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A polynomial equation is an equation that involves a polynomial expression, which is a sum of terms, each consisting of a variable raised to a non-negative integer power and multiplied by a coefficient.

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What does it mean to factor a polynomial?

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Factoring a polynomial means expressing it as a product of its factors, which are polynomials of lower degrees.

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What is the zero of a polynomial?

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A zero of a polynomial is a value of the variable that makes the polynomial equal to zero.

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How do you determine if a binomial is a factor of a polynomial?

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To determine if a binomial is a factor of a polynomial, you can use polynomial long division or synthetic division. If the remainder is zero, then the binomial is a factor.

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What is the Factor Theorem?

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The Factor Theorem states that if f(c) = 0 for a polynomial f(x), then (x - c) is a factor of f(x).

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What is the process of factoring by grouping?

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Factoring by grouping involves rearranging the terms of a polynomial and grouping them in pairs to factor out common factors.

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How do you solve a polynomial equation by factoring?

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To solve a polynomial equation by factoring, first factor the polynomial, then set each factor equal to zero and solve for the variable.

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What is a cubic polynomial?

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A cubic polynomial is a polynomial of degree three, which can be expressed in the form f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants.

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What is the significance of finding all zeros of a polynomial?

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Finding all zeros of a polynomial helps in understanding its graph, as the zeros represent the x-intercepts where the polynomial crosses the x-axis.

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What is synthetic division?

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Synthetic division is a simplified form of polynomial long division that is used to divide a polynomial by a linear factor.