Problem 7 Use the guess and check method t... [FREE SOLUTION]

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Elementary Algebra

Laura Bracken, Ed Miller

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Chapter 6: Problem 7

Use the guess and check method to factor. Identify any prime polynomials. $$ z^{2}-7 z-18 $$

Short Answer

Expert verified

$(z + 2)(z - 9)$

Step by step solution

Understand the problem

To factor the polynomial $z^2 - 7z - 18$, look for two numbers that multiply to -18 (the constant term) and add up to -7 (the coefficient of the linear term).

List factor pairs

List the pairs of factors of -18: (1, -18), (-1, 18), (2, -9), (-2, 9), (3, -6), (-3, 6).

Identify correct pair

Identify the pair that adds up to -7. Checking each pair, (2, -9) works because 2 + (-9) = -7.

Write the factors

Write the polynomial as a product of two binomials: $(z + 2)(z - 9)$.

Verify the factorization

Expand $(z + 2)(z - 9)$ to check if it equals $z^2 - 7z - 18$: $(z + 2)(z - 9) = z^2 - 9z + 2z - 18 = z^2 - 7z - 18$. The factorization is correct.

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Key Concepts

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

guess and check method

The guess and check method is a straightforward approach used to factor polynomials. It involves listing all possible factor pairs of the constant term and checking which pairs sum up to the coefficient of the linear term. The method can be a bit tedious, but it ensures that you find the correct factors. By systematically guessing pairs and checking their sums, you can identify the correct pair of numbers to use in your factorization.

factor pairs

When factoring polynomials, the factor pairs of the constant term play a crucial role. These pairs are combinations of numbers that multiply to give the constant term. For example, in the polynomial $z^2 - 7z - 18$, the constant term is -18. Writing down the pairs of factors of -18 gives us:

(1, -18)
(-1, 18)
(2, -9)
(-2, 9)
(3, -6)
(-3, 6)

It's important to consider both negative and positive pairs, as one of them will add up to match our desired coefficient for the linear term.

verifying factorization

Once you've factored the polynomial, it's essential to verify that the factorization is correct. To do this, expand the binomials and check if the result matches the original polynomial. For $z^2 - 7z - 18$, after identifying $ (z + 2)(z - 9)$ as the factors, you should expand it to ensure accuracy.
So, we expand:
\[(z + 2)(z - 9) = z^2 - 9z + 2z - 18 = z^2 - 7z - 18\]
Since the resulting polynomial matches the original, the factorization is verified as correct. This crucial step helps in confirming that no mistakes were made during the process.

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91影视

Use the guess and check method to factor. Identify any prime polynomials. $$ z^{2}-7 z-18 $$

Short Answer

Step by step solution

Understand the problem

List factor pairs

Identify correct pair

Write the factors

Verify the factorization

Key Concepts

guess and check method

factor pairs

verifying factorization

One App. One Place for Learning.

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