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Chapter 5: Problem 174

Factor each polynomial using the trial-and-error method. $$ 2 w^{2}-23 w+11 $$

Short Answer

Expert verified
The factorization of \(2w^2-23w+11\) is \((2w-1)(w-11)\).

Step by step solution

01

- Identify the polynomial

Identify the given polynomial, which is: \(2w^{2}-23w+11\)
02

- Finding factor pairs of the constant term

List the factor pairs of the constant term 11. They are: \((1, 11)\) and \((-1, -11)\).
03

- Setting up trial pairs

Set up potential binomials. For this polynomial, potential binomials are: \((2w + a)(w + b)\). Focus on when product terms have \(2w^2\) and when their coefficients add to -23.
04

- Check products sum

Check whether the terms of the product give the correct middle coefficient. Use: \((2w - 1)(w - 11)\). Calculate inner and outer terms sum: \( (2w \cdot -11) + (w \cdot -1) = -22w - 1w = -23w \)
05

- Verify factorization

Expand to verify: \((2w - 1)(w - 11) = 2w^2 - 22w - 1w + 11 = 2w^2 - 23w + 11\) The factorization is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

trial-and-error method
The trial-and-error method is a commonly used technique for factoring polynomials. It involves testing various combinations of factor pairs until the correct one is found. To apply this method, we start by identifying the polynomial, like the one we have: \(2w^{2}-23w+11\).

We look for pairs of binomials whose product recreates the original polynomial. This might sound complex, but it's all about patience and testing different possibilities.

In our example, we try out different combinations to see which ones, when multiplied together, result in \(2w^{2}-23w+11\).
polynomial factorization
Polynomial factorization involves expressing a polynomial as a product of simpler polynomials. For example, if we have \(2w^{2}-23w+11\), we want to rewrite it as a product of two binomials.

We start by listing factor pairs of the constant term, which in this case is 11: (1, 11) and (-1, -11). Next, we set up our potential binomials, aiming to find a pair that multiplies back to the original polynomial.

By the trial-and-error method, we find that \((2w - 1)(w - 11)\) is the correct factorization because its expanded form, \(2w^2 - 22w - 1w + 11\), gives us our original polynomial: \(2w^2 - 23w + 11\).
binomials
Binomials are polynomials with two terms. In the context of factoring, we aim to express a polynomial like \(2w^{2}-23w+11\) as the product of two binomials.

This is done by finding binomials whose product matches the quadratic polynomial. For our example, the binomials \((2w - 1)\) and \((w - 11)\) multiply to give us back the original polynomial.

Checking the factorization helps ensure accuracy. We expand \((2w - 1)(w - 11)\) to get \(2w^2 - 22w - 1w + 11\), which simplifies to \(2w^2 - 23w + 11\), confirming our binomials are correctly identified.

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