Polynomials & Their Zeros

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What is a polynomial?

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A polynomial is a mathematical expression consisting of variables (also called indeterminates) raised to non-negative integer powers and coefficients. For example, 2x^3 - 4x^2 + 3x - 5 is a polynomial.

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What are the zeros of a polynomial?

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

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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. This can involve factoring, using the quadratic formula, or other algebraic methods.

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

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If 'r' is a zero of a polynomial, then (x - r) is a factor of that polynomial. This means that the polynomial can be expressed as a product of its factors.

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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 the polynomial 3x^4 + 2x^3 - x + 7, the degree is 4.

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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 factor of the form (x - r). It is faster than long division and is used to find the quotient and remainder.

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What is the Remainder Theorem?

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The Remainder Theorem states that if a polynomial f(x) is divided by (x - r), the remainder of that division is f(r). This helps in finding zeros of the polynomial.

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What is the Factor Theorem?

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The Factor Theorem states that (x - r) is a factor of the polynomial f(x) if and only if f(r) = 0. This is used to determine if a given value is a zero.

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How do you build a polynomial from its zeros?

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To build a polynomial from its zeros, use the fact that if r1, r2, ..., rn are the zeros, the polynomial can be expressed as f(x) = k(x - r1)(x - r2)...(x - rn), where k is a non-zero constant.

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What is the standard form of a polynomial?

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The standard form of a polynomial is written in descending order of the degree of its terms. For example, 4x^3 - 2x^2 + x - 5 is in standard form.