IM3: Polynomial Functions, Factors, and Division 2018

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What is a polynomial function?

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A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is P(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n ≠ 0.

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What does it mean for a polynomial to be divided by a linear factor?

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When a polynomial P(x) is divided by a linear factor (x - c), the result is a quotient polynomial Q(x) and a remainder R, such that P(x) = (x - c)Q(x) + R.

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What is the Remainder Theorem?

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The Remainder Theorem states that if a polynomial P(x) is divided by (x - c), the remainder of this division is P(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 P(x) if and only if P(c) = 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 the variable. The solutions are the x-values where the polynomial intersects the x-axis.

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What is the significance of the degree of a polynomial?

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The degree of a polynomial indicates the highest power of the variable in the polynomial. It determines the polynomial's end behavior and the maximum number of roots it can have.

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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 divisor of the form (x - c). It is faster than long division and involves using coefficients.

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What is a quadratic equation?

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A quadratic equation is a polynomial equation of degree 2, typically in the form ax^2 + bx + c = 0, where a ≠ 0.

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What are the roots of a polynomial?

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The roots of a polynomial are the values of x that make the polynomial equal to zero. They are also known as the solutions or zeros of the polynomial.

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What is the relationship between the roots and factors of a polynomial?

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If r is a root of the polynomial P(x), then (x - r) is a factor of P(x).