Chapter 0: Problem 34
Factor the trinomial.\(2 x^{2}-x-1\)
Short Answer
Expert verified
The factored form of the trinomial \(2 x^{2}-x-1\) is \((2x + 1)(x - 1)\).
Step by step solution
01
Identify the coefficients
The coefficients for our trinomial \(2 x^{2}-x-1\) are a = 2, b = -1 and c = -1.
02
Listing the Factors
Since we are looking for two numbers that add to b and multiply to a*c, we list the possible factors of -2 (product of a and c). The factors pairs are (-1,2), and (1,-2).
03
Identify correct pair
The pair that adds to -1 (our b value) is (1, -2). The first number, 1, is our factor 'u' and the second number, -2, is our factor 'v'.
04
Rewrite the trinomial
Rewrite the trinomial by breaking up the middle term, -x, into terms using 'u' and 'v'. This gives us: \(2 x^{2} + (u)*x + (v)*x - 1\), substituting for u and v, we get \(2 x^{2} + x - 2x - 1\).
05
Factor by grouping
Group the first two terms and the last two terms, and then factor out common factors: \(x(2x + 1) - 1(2x + 1)\). It is noticeable that both groups contain the same binomial factor, \(2x + 1\). This allows us to further factor the expression.
06
Final factorization
We can now factor the binomial term out of the two groups, giving us the final factored form: \((2x + 1)(x - 1)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Factorization
Polynomial factorization is a mathematical technique used to break down a polynomial into simpler components, known as factors, which when multiplied together give back the original polynomial. In the context of our exercise, we tackled the factorization of a trinomial: a polynomial with three terms.
To factor the given trinomial, \(2x^2-x-1\), we first identify the coefficients, and then we look for two numbers that when added give the middle coefficient, -1, and when multiplied provide the product of the first and last coefficients, in this case, -2 (from \(2x^2\) and \(1\)).
The beauty of polynomial factorization lies in its ability to simplify complex expressions and solve equations more efficiently. Once factorized, the roots or solutions of these polynomials can be discovered, making this technique highly valuable across various fields of mathematics and science.
To factor the given trinomial, \(2x^2-x-1\), we first identify the coefficients, and then we look for two numbers that when added give the middle coefficient, -1, and when multiplied provide the product of the first and last coefficients, in this case, -2 (from \(2x^2\) and \(1\)).
The beauty of polynomial factorization lies in its ability to simplify complex expressions and solve equations more efficiently. Once factorized, the roots or solutions of these polynomials can be discovered, making this technique highly valuable across various fields of mathematics and science.
Binomial Factors
Binomial factors are expressions that contain two terms, and they play a significant role in the process of factoring polynomials. In our exercise, the trinomial is eventually broken down into two binomial factors.
During factorization, we often seek to rewrite the polynomial so that it contains a common binomial factor in its grouped terms. Here, after reorganizing the trinomial and factoring by grouping, we discovered that the two groups shared a binomial factor, \(2x + 1\). Identifying common binomial factors is crucial as it allows us to pull these out and further reduce the polynomial into its simplest form.
Understanding how to manipulate and recognize binomial factors equips students to handle more complex factorizations involving polynomials of higher degrees.
During factorization, we often seek to rewrite the polynomial so that it contains a common binomial factor in its grouped terms. Here, after reorganizing the trinomial and factoring by grouping, we discovered that the two groups shared a binomial factor, \(2x + 1\). Identifying common binomial factors is crucial as it allows us to pull these out and further reduce the polynomial into its simplest form.
Understanding how to manipulate and recognize binomial factors equips students to handle more complex factorizations involving polynomials of higher degrees.
Factor by Grouping
Factor by grouping is a strategic method used in polynomial factorization, especially when dealing with four-term polynomials or in certain cases of trinomials like ours. It involves rearranging the terms into groups that have a common factor, making it easier to simplify.
In our example \(2x^2 + x - 2x - 1\), we grouped the terms into \(2x^2 + x\) and \( - 2x - 1\), then we factored out the greatest common factor from each group. Noticeably, this resulted in both groups having a common binomial factor \(2x + 1\). Then, by factoring out this common binomial factor, we reached the final factored form of the trinomial: \(2x + 1)(x - 1)\).
The technique of grouping provides an orderly path to simplify expressions that may initially appear complex. It's an essential tool for students, as it reveals the underlying structure of the polynomial and highlights the relationships between its terms.
In our example \(2x^2 + x - 2x - 1\), we grouped the terms into \(2x^2 + x\) and \( - 2x - 1\), then we factored out the greatest common factor from each group. Noticeably, this resulted in both groups having a common binomial factor \(2x + 1\). Then, by factoring out this common binomial factor, we reached the final factored form of the trinomial: \(2x + 1)(x - 1)\).
The technique of grouping provides an orderly path to simplify expressions that may initially appear complex. It's an essential tool for students, as it reveals the underlying structure of the polynomial and highlights the relationships between its terms.
