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Chapter 5: Problem 132

Factor each polynomial completely. $$z^{2}-14 z+49$$

Short Answer

Expert verified
The factored form is \( (z-7)^2 \).

Step by step solution

01

Identify the Polynomial Type

Recognize that the given polynomial is a quadratic polynomial of the form \( ax^{2} + bx + c \). Here, \( a = 1 \), \( b = -14 \), and \( c = 49 \).
02

Look for Patterns

Check if the quadratic polynomial can be factored into two binomials. Note that this polynomial may fit the pattern of a perfect square trinomial \( (x - d)^2 \).
03

Check for Perfect Square

Confirm whether the terms form a perfect square trinomial by comparing both the middle term and the constant term. Here, \( z^{2} \ z^{2} \), and \( 49 \ (7)^2 \). The middle term \(-14z\) fits because \( 2 \times 7 \times z = 14z \).
04

Write as a Square Binomial

Since the polynomial fits the perfect square trinomial pattern, rewrite it as \( (z - 7)^{2} \).
05

Verify Solution

To verify, expand \( (z-7)^2 \) back to \( z^2 - 14z + 49 \) to ensure accuracy.

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Key Concepts

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

quadratic polynomial
A quadratic polynomial is a polynomial of degree 2. This means the highest power of the variable (let's call it z) is 2. Quadratic polynomials have the general form:
  • \( a z^2 + b z + c \),
  • where a, b, and c are constants and \( a eq 0 \).

For example, in the problem we have:
  • \( z^2 - 14z + 49 \)
  • which is our quadratic polynomial with \( a = 1 \), \( b = -14 \), and \( c = 49 \).

Understanding how to identify and work with quadratic polynomials is crucial because they appear so often in algebra. We solve them by factoring, using the quadratic formula, or by completing the square.
perfect square trinomial
A perfect square trinomial is a special type of quadratic polynomial. It takes the form: \( (x + d)^2 \) or \( (x - d)^2 \). When expanded, it results in:
  • \( x^2 + 2dx + d^2 \) or \( x^2 - 2dx + d^2 \).

In the given exercise, we need to recognize the pattern of a perfect square trinomial. The polynomial
  • \( z^2 - 14z + 49 \),
fits this pattern because:
  • \( z^2 = (z)^2 \)
  • \( 49 = (7)^2 \)
  • and
    \( -14z = 2 \times -7 \times z \).

Thus, we identify it as a perfect square trinomial, making it easier to factor into a binomial squared.
binomial factorization
Binomial factorization involves rewriting the polynomial as the product of two binomials. For perfect square trinomials, this simplifies further. Our polynomial
  • \( z^2 - 14z + 49 \),
can be rewritten as
  • \( (z - 7)^2 \)
because:
  • \( z - 7 \times z - 7 \)

    • When factored, we group the terms:
    • \( (z - 7) \times (z - 7) \)
    This is beneficial because it simplifies expressions and equations, making them easier to solve. Always remember to verify by expanding the binomials to ensure accuracy, which in this case confirms:
    • \( (z - 7)^2 = z^2 - 14z + 49 \).

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Most popular questions from this chapter

Factor each polynomial by grouping. $$ n^{2} b+3 b+15+5 n^{2} $$

Factor each polynomial using the trial-and-error method. $$ 2 w^{2}+21 w-11 $$

Factor each polynomial and explain how you decided which method to use. a) \(x^{2}+10 x+25\) b) \(x^{2}-10 x+25\) c) \(x^{2}+26 x+25\) d) \(x^{2}-25\) e) \(x^{2}+25\)

Factor each polynomial using the trial-and-error method. $$ 3 x^{2}-17 x-6 $$

Factor each polynomial using the trial-and-error method. $$ b^{2}-4 b+3 $$
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