Problem 8 How do you check the factorizati... [FREE SOLUTION]

Chapter 7: Problem 8

How do you check the factorization of a polynomial?

Short Answer

Expert verified

To check the factorization of a polynomial, multiply the factors using the distributive property, then simplify and compare the resulting polynomial with the original one. For example, to check the factorization of \(P(x) = 2x^2 - 5x - 3\) as \((2x +1)(x-3)\), multiply the factors and simplify: \((2x + 1)(x - 3) = 2x^2 - 6x + x - 3 = 2x^2 - 5x - 3\). Since the resulting polynomial matches the original, the factorization is correct.

Step by step solution

Understand the given polynomial and its factors

A polynomial is an algebraic expression consisting of variables and coefficients, formed by the addition or subtraction of monomials. A polynomial's factorization is dividing it into multiple simpler expressions that, when multiplied together, give the original polynomial. Consider a polynomial P(x) and its factorization given as a product of simpler polynomials F(x) and G(x). It is denoted as follows: \[P(x) = F(x) \cdot G(x)\] The goal is to check that the multiplication of F(x) and G(x) results in the original polynomial P(x).

Multiply the factors using the distributive property

Using the distributive property of multiplication, we will multiply the factors in the given factorization expression. When multiplying polynomials, each term in the first polynomial must be multiplied by each term in the second polynomial, and then the results should be combined and simplified.

Check if the product of factors simplifies to the original polynomial

Once the factor multiplication is completed, compare the resulting polynomial with the original polynomial provided. If the resulting polynomial matches the original polynomial, then the factorization is correct. Otherwise, the factorization is incorrect.

Example

Let's consider a polynomial and its given factorization as an example: \[P(x) = 2x^2 - 5x - 3\] with the factorization \[(2x+1)(x-3)\]. Step 1: We understand that we have a polynomial P(x) and its factors F(x) and G(x) are (2x + 1) and (x - 3), respectively. Step 2: Multiply the two factors using the distributive property: \[(2x + 1)(x - 3) = 2x(x) + 2x(-3) + 1(x) + 1(-3)\] Step 3: Combine and simplify the terms in the resulting product: \[2x^2 - 6x + x - 3 = 2x^2 - 5x - 3\] The product of the factors matches the original polynomial P(x), so the factorization is correct.

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91影视

How do you check the factorization of a polynomial?

Short Answer

Step by step solution

Understand the given polynomial and its factors

Multiply the factors using the distributive property

Check if the product of factors simplifies to the original polynomial

Example

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