Factoring by grouping

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What is factoring by grouping?

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Factoring by grouping is a method used to factor polynomials by grouping terms with common factors and then factoring out the common binomial.

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What is a common factor?

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A common factor is a number or expression that divides two or more numbers or expressions without leaving a remainder.

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Group the terms: (20p^3 - 25p^2) + (4p - 5) = 5p^2(4p - 5) + 1(4p - 5) = (5p^2 + 1)(4p - 5).

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What is the first step in factoring by grouping?

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The first step is to group the terms in pairs that have common factors.

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Group the terms: (x^3 - 5x^2) + (5x - 25) = x^2(x - 5) + 5(x - 5) = (x^2 + 5)(x - 5).

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How do you check if a binomial is a factor of a polynomial?

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You can check by performing polynomial long division or synthetic division. If the remainder is zero, it is a factor.

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Group the terms: (x^3 - 4x^2) + (-4x + 16) = x^2(x - 4) - 4(x - 4) = (x - 4)(x^2 - 4).

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What is the significance of the binomial in factoring by grouping?

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The binomial represents a common factor that can be factored out from the grouped terms, simplifying the polynomial.