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Apply the distributive property to polynomials

The distributive property preserves equivalence by requiring each term of one factor to multiply every term of the other, including variables, negative or fractional coefficients, and nonnegative integer exponents; resulting like terms may then be combined. This applies to expanding products of a monomial and a polynomial or of two manageable polynomials, without extending to abstract polynomial-ring generalizations or advanced identities beyond these expressions.

Detailed Explanation: Apply the distributive property to polynomials

To expand a product of polynomials, multiply every term in the first factor by every term in the second factor. Then simplify by combining like terms.

Example:

(2x−3)(x+4)(2x-3)(x+4)

First, distribute 2x2x to both terms in the second factor:

2x(x+4)=2xâ‹…x+2xâ‹…42x(x+4)=2x\cdot x+2x\cdot 4

Now distribute −3-3 to both terms:

−3(x+4)=−3⋅x+(−3)⋅4-3(x+4)=-3\cdot x+(-3)\cdot 4

Put all the products together:

(2x−3)(x+4)=2x2+8x−3x−12(2x-3)(x+4)=2x^2+8x-3x-12

Use the exponent rule x⋅x=x2x\cdot x=x^2. Then combine the like terms 8x8x and −3x-3x:

2x2+8x−3x−12=2x2+5x−122x^2+8x-3x-12=2x^2+5x-12

Therefore,

(2x−3)(x+4)=2x2+5x−12\boxed{(2x-3)(x+4)=2x^2+5x-12}

Be sure that each term is multiplied by every term in the other factor, and keep track of negative signs.

Learn by doing: Apply the distributive property to polynomials

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