Solving Polynomial Equations

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What are the zeros of a polynomial function?

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The zeros of a polynomial function are the values of x for which the function equals zero. They are also known as the roots or x-intercepts of the function.

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How do you write a polynomial in standard form?

Back

A polynomial is written in standard form when its terms are arranged in descending order of the degree of each term, with the highest degree term first.

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What does it mean for a function to cross the x-axis?

Back

When a function crosses the x-axis, it means that the output (y-value) is zero at that point, indicating that the corresponding x-value is a zero of the function.

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What is the factored form of a polynomial?

Back

The factored form of a polynomial expresses it as a product of its factors, which can include linear factors and irreducible quadratic factors.

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What is the difference between a linear factor and a quadratic factor?

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A linear factor is a polynomial of degree 1 (e.g., (x - a)), while a quadratic factor is a polynomial of degree 2 (e.g., (x^2 + bx + c)).

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How do you factor a difference of squares?

Back

A difference of squares can be factored using the formula a^2 - b^2 = (a - b)(a + b).

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What is synthetic division, and when is it used?

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Synthetic division is a simplified form of polynomial long division used to divide a polynomial by a linear factor of the form (x - c).

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What is the Remainder Theorem?

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The Remainder Theorem states that when a polynomial f(x) is divided by (x - c), the remainder of that division is equal to f(c).

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What is the Factor Theorem?

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The Factor Theorem states that (x - c) is a factor of the polynomial f(x) if and only if f(c) = 0.

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How do you find the roots of a polynomial equation?

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To find the roots of a polynomial equation, set the polynomial equal to zero and solve for x using factoring, synthetic division, or the quadratic formula.