Polynomial Factoring and Solving

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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, 4x^3 + 5x^2 - 16x is a polynomial.

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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, which are polynomials of lower degrees. For example, the polynomial x^2 - 9 can be factored as (x - 3)(x + 3).

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What is the zero-product property?

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The zero-product property states that if the product of two or more factors is zero, at least one of the factors must be zero. This is used to solve polynomial equations.

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How do you solve a polynomial equation?

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To solve a polynomial equation, you can set the equation to zero and factor it, then use the zero-product property to find the values of the variable that satisfy the equation.

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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 4x^3 + 5x^2 - 16x, the degree is 3.

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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, b, and c are constants.

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What is the difference of squares?

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The difference of squares is a special factoring pattern that states a^2 - b^2 = (a - b)(a + b).

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What is the sum of cubes?

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The sum of cubes is a factoring pattern that states a^3 + b^3 = (a + b)(a^2 - ab + b^2).

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What is the product of cubes?

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The product of cubes is a factoring pattern that states a^3 - b^3 = (a - b)(a^2 + ab + b^2).

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How do you factor a polynomial with a common factor?

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To factor a polynomial with a common factor, identify the greatest common factor (GCF) of all terms and factor it out. For example, in 4x^3 + 8x^2, the GCF is 4x^2, so it can be factored as 4x^2(x + 2).