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Chapter 0: Problem 29

$$ \text { Factor the polynomials in Exercises } \text { . } $$ $$ x^{2}-14 x+49 $$

Short Answer

Expert verified
\((x - 7)^2\)

Step by step solution

01

Identify the polynomial structure

Examine the given polynomial, which is in the form of a quadratic equation: \(x^{2} - 14x + 49\).
02

Recognize perfect square trinomial

Notice that the polynomial can be written as a perfect square trinomial: \(x^{2} - 14x + 49 = (a - b)^2\).
03

Identify the terms

Identify the values for \(a\) and \(b\). Recognize that \(a = x\) and \(b = 7\), since \(7^2 = 49\) and \(2 \times x \times 7 = 14x\).
04

Rewrite the polynomial

Rewrite the polynomial as the square of a binomial: \((x - 7)^2\).
05

Verify the factorization

Expand the binomial to ensure the factorization is correct: \((x - 7)(x - 7) = x^{2} - 14x + 49\).

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Key Concepts

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

Quadratic Equation
A quadratic equation is a type of polynomial equation of degree 2. This means its highest exponent is 2. The general form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a, b,\) and \(c\) are constants and \(a eq 0\). Quadratic equations can be solved using various methods:

* Factoring
* Using the quadratic formula
* Completing the square

In our exercise, we are dealing with the quadratic polynomial \(x^2 - 14x + 49\). To solve it by factoring, we recognize it as a perfect square trinomial.
Perfect Square Trinomial
A perfect square trinomial is a special form of a quadratic polynomial. It can be written as the square of a binomial. The general forms are \(a^2 + 2ab + b^2 = (a + b)^2\) and \(a^2 - 2ab + b^2 = (a - b)^2\).

In our problem, \(x^2 - 14x + 49\) is a perfect square trinomial. Here's why:

* \(x^2\) is the square of \(x\)
* \(-14x\) is twice the product of \(x\) and 7 (so, 2 times \(x\) times 7)
* \(+49\) is the square of 7

Therefore, \(x^2 - 14x + 49\) can be written as \((x - 7)^2\). This makes it easier to handle and solve.
Binomial Expansion
Binomial expansion is a process where you expand an expression that is raised to a power. The binomial theorem provides a formula to do this efficiently.

For example, when expanding \((a + b)^2\), we use:

\((a + b)^2 = a^2 + 2ab + b^2\)

In our exercise, we rewrote the trinomial \(x^2 - 14x + 49\) as \((x - 7)^2\). This is because:
  • \(a = x\) and \(b = 7\)
To check the factorization, we expanded \( (x - 7)^2 \) back to its original form:

\( (x - 7)(x - 7) = x^2 - 14x + 49\)

Perfect square trinomials are handy because they simplify polynomials into squares of binomials, making them easier to solve or factor.

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Most popular questions from this chapter

In Exercises 41-70, use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. \((x y)^{6}\)

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. \(\left(\frac{x}{y}\right)^{-2}\)

A college student earns income by typing term papers on a computer, which she leases (along with a printer). The student charges $$\$ 4$$ per page for her work, and she estimates that her monthly cost when typing \(x\) pages is \(C(x)=.10 x+75\) dollars. (a) What is the student's profit if she types 100 pages in 1 month? (b) Determine the change in profit when the typing business rises from 100 to 101 pages per month.

The expressions in Exercises 83-88 may be factored as shown. Find the missing factors. \(2 x^{2 / 3}-x^{-1 / 3}=x^{-1 / 3}(\quad)\)

During the first \(\frac{1}{2}\) hour, the employees of a machine shop prepare the work area for the day's work. After that, they turn out 10 precision machine parts per hour, so the output after \(t\) hours is \(f(t)\) machine parts, where \(f(t)=10\left(t-\frac{1}{2}\right)=10 t-5, \frac{1}{2} \leq t \leq 8\). The total cost of producing \(x\) machine parts is \(C(x)\) dollars, where \(C(x)=.1 x^{2}+25 x+200\). (a) Express the total cost as a (composite) function of \(t\). (b) What is the cost of the first 4 hours of operation?
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