Solving Polynomials Review

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What is the standard form of a polynomial?

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The standard form of a polynomial is expressed as a sum of terms in descending order of their degrees, such as: $$a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$$ where $$a_n$$ are coefficients and $$n$$ is a non-negative integer.

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How do you find the roots of a quadratic equation?

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To find the roots of a quadratic equation in the form $$ax^2 + bx + c = 0$$, you can use the quadratic formula: $$x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}$$.

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What is the degree of a polynomial?

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The degree of a polynomial is the highest power of the variable in the polynomial. For example, in $$7x^3 - 8x^2 + 9$$, the degree is 3.

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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. For example, $$4x^2 - 8x$$ can be factored as $$4x(x - 2)$$.

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What is the difference between a monomial and a polynomial?

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A monomial is a single term (e.g., $$3x^2$$), while a polynomial is a sum of two or more monomials (e.g., $$3x^2 + 2x + 1$$).

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How do you divide a polynomial by a monomial?

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To divide a polynomial by a monomial, divide each term of the polynomial by the monomial. For example, $$\frac{{15x^9 - 8x^3}}{{x^3}} = 15x^6 - 8$$.

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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 is $$f(c)$$.

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What is synthetic division?

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Synthetic division is a simplified method of dividing a polynomial by a linear binomial of the form $$x - c$$. It uses coefficients and is faster than long division.

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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 solve a polynomial equation?

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To solve a polynomial equation, set the polynomial equal to zero and find the values of the variable that satisfy the equation, often using factoring, the quadratic formula, or numerical methods.