Factoring Polynomials

Front

What is factoring in algebra?

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Factoring is the process of breaking down an expression into a product of simpler expressions, or factors, that when multiplied together give the original expression.

Front

What is a polynomial?

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A polynomial is a mathematical expression that consists of variables, coefficients, and non-negative integer exponents, combined using addition, subtraction, and multiplication.

Front

What is the standard form of a quadratic polynomial?

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The standard form of a quadratic polynomial is ax² + bx + c, where a, b, and c are constants and a ≠ 0.

Front

What are the steps to factor a quadratic polynomial?

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1. Identify the coefficients a, b, and c. 2. Find two numbers that multiply to ac and add to b. 3. Rewrite the middle term using these two numbers. 4. Factor by grouping.

Front

What is the difference between a prime polynomial and a composite polynomial?

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A prime polynomial cannot be factored into simpler polynomials with integer coefficients, while a composite polynomial can be factored into two or more non-trivial polynomials.

Front

What is the significance of the zero product property in factoring?

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The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. This is used to find the roots of polynomial equations.

Front

How do you factor a polynomial with a leading coefficient greater than 1?

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To factor a polynomial with a leading coefficient greater than 1, find two numbers that multiply to the product of the leading coefficient and the constant term, and add to the middle coefficient. Then, use these numbers to split the middle term and factor by grouping.

Front

What is the role of the discriminant in factoring quadratic polynomials?

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The discriminant (b² - 4ac) determines the nature of the roots of a quadratic polynomial. If it is positive, there are two distinct real roots; if zero, one real root; and if negative, no real roots.

Front

What is the factored form of the polynomial k² + 7k + 10?

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The factored form is (k + 2)(k + 5).

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What is the factored form of the polynomial 3a² + 4a + 1?

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The factored form is (3a + 1)(a + 1).