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Chapter 0: Problem 50

Factor each polynomial. $$ x^{2}-17 x+16 $$

Short Answer

Expert verified
The factored form is \( (x - 1)(x - 16) \).

Step by step solution

01

Identify Coefficients

Look at the polynomial equation and identify the coefficients for each term. For the given polynomial \[ x^2 - 17x + 16 \] the coefficients are: - Leading coefficient (next to \(x^2\)): 1 - Middle coefficient (next to \(x\)): -17 - Constant term: 16
02

Find the Product and Sum

Next, find two numbers that multiply to give the constant term (16) and add to give the middle coefficient (-17). Look for pairs of factors of 16: 1, 16 2, 8 4, 4 Among these, the pair that sums up to -17 is -1 and -16, because \[-1 + (-16) = -17\] and \[-1 \times (-16) = 16\].
03

Write in Factored Form

Use the numbers identified in the previous step to write the polynomial in its factored form: \[ x^2 - 17x + 16 = (x - 1)(x - 16) \]Check the factorization to confirm: \[(x - 1)(x - 16) = x^2 - 16x - x + 16 = x^2 - 17x + 16\].

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Key Concepts

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

coefficients identification
To start factoring a polynomial, you first need to identify its coefficients. Coefficients are the numbers in front of the variables. Let's break down the given polynomial \( x^2 - 17x + 16 \). Here are the coefficients you need to know:
  • Leading coefficient (next to \( x^2 \)): \( 1 \)
  • Middle coefficient (next to \( x \)): \( -17 \)
  • Constant term: \( 16 \)
Identifying coefficients correctly sets the stage for the next steps. Always write them down so you won't make mistakes later.
product and sum method
Once you know your coefficients, the next step is to use the product and sum method. This method helps you find two numbers that both multiply to the constant term and add up to the middle coefficient.
Start by listing pairs of factors of the constant term, which is 16 in our case. These pairs include:
  • 1, 16
  • 2, 8
  • 4, 4
Now, we need to find which of these pairs also add up to the middle coefficient, which is -17.
The correct pair is \( -1 \) and \( -16 \):
  • \( -1 + (-16) = -17 \)
  • \( -1 \times (-16) = 16 \)
Finding the right pair is crucial because it simplifies the polynomial into factors next.
factored form
Now that we have the numbers \( -1 \) and \( -16 \) from the product and sum method, we can write the polynomial in its factored form. This gives us: \[ x^2 - 17x + 16 = (x - 1)(x - 16) \]
To make sure this is correct, we can do a quick check:
  • First, expand \( (x - 1)(x - 16) \): \[ (x - 1)(x - 16) = x^2 - 16x - x + 16 \]
  • Combine like terms to get back to the original polynomial: \[ x^2 - 17x + 16 \]
The factored form lets you see the roots of the polynomial more easily. It's crucial for solving polynomial equations and for simplifying expressions in algebra. Factoring is a key skill for mastering higher-level math concepts.

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