Adv Algebra Practice Flashcard: Factor and Solve Polynomials

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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 the degree of each term, typically written as: f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n ≠ 0.

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How do you find the zeros of a polynomial?

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To find the zeros of a polynomial, set the polynomial equal to zero and solve for x. This can involve factoring, using the quadratic formula, or applying synthetic division.

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What does it mean for a number to be a zero of a polynomial?

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A number is a zero of a polynomial if substituting that number into the polynomial results in a value of zero.

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What is the relationship between the zeros of a polynomial and its factors?

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The zeros of a polynomial correspond to the values of x that make the polynomial equal to zero, and each zero (r) can be expressed as a factor of the form (x - r).

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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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How do you factor a polynomial completely?

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To factor a polynomial completely, identify and factor out the greatest common factor (GCF), then apply techniques such as grouping, synthetic division, or special factoring formulas (e.g., difference of squares, perfect square trinomials).

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What is the difference of cubes formula?

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The difference of cubes formula is a^3 - b^3 = (a - b)(a^2 + ab + b^2).

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What is the sum of cubes formula?

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The sum of cubes formula is a^3 + b^3 = (a + b)(a^2 - ab + b^2).

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How do you factor a polynomial of the form ax^3 + b?

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To factor a polynomial of the form ax^3 + b, use the sum of cubes formula if b is positive, or the difference of cubes formula if b is negative.

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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 of the form (x - c).