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Chapter 3: Problem 76

Write a polynomial \(f(x)\) that meets the given conditions. Answers may vary. (See Example 10 ) Degree 2 polynomial with zeros \(5 \sqrt{2}\) and \(-5 \sqrt{2}\).

Short Answer

Expert verified
f(x) = x^2 - 50

Step by step solution

01

Identify the Zeros

The given zeros are \(5 \, \sqrt{2}\) and \(-5 \, \sqrt{2}\).
02

Form Factors

Based on the zeros, the factors of the polynomial are \((x - 5 \, \sqrt{2})\) and \((x + 5 \, \sqrt{2})\).
03

Multiply the Factors

Multiply the factors to form the polynomial: \[ (x - 5 \, \sqrt{2})(x + 5 \, \sqrt{2}) \]
04

Expand

Apply the difference of squares formula: \[ (a - b)(a + b) = a^2 - b^2 \] So, \[ (x - 5 \, \sqrt{2})(x + 5 \, \sqrt{2}) = x^2 - (5 \, \sqrt{2})^2 \]
05

Simplify

Simplify the polynomial: \[ x^2 - (5 \, \sqrt{2})^2 = x^2 - 50 \]

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Key Concepts

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

Degree 2 Polynomial
A degree 2 polynomial, also known as a quadratic polynomial, is a polynomial of the form \( ax^2 + bx + c \) where * \(a\), \(b\), and \(c\) are constants, * \(a eq 0\).The highest exponent in a degree 2 polynomial is 2.
For example, in the exercise, our polynomial is initially given by its factors, which when multiplied, result in a quadratic equation, confirming its degree 2 status.
Zeros of Polynomial
The zeros (or roots) of a polynomial are the values of \(x\) that make the polynomial equal to zero. In other words, if \(f(x)\) is a polynomial, the values such that \(f(x) = 0\) are its zeros.

In the given problem, the zeros are \(5\sqrt{2}\) and \(-5\sqrt{2}\). What this means is that when we substitute these values for \(x\) in our polynomial \(f(x)\), the result will be zero.
Difference of Squares
The difference of squares formula is a handy tool for expanding and simplifying expressions of the form \((a - b)(a + b)\). The formula states that: \[ (a - b)(a + b) = a^2 - b^2 \]In our exercise, the factors of the polynomial \((x - 5\sqrt{2})(x + 5\sqrt{2})\) is a perfect example.

Applying the difference of squares, we set \(a = x\) and \(b = 5\sqrt{2}\), leading to: \[ x^2 - (5\sqrt{2})^2 \].
Expansion and Simplification
Expansion is the process of multiplying out the factors to form a polynomial. Simplification is the process of reducing the polynomial to its simplest form.
In the exercise, we start by expanding \((x - 5\sqrt{2})(x + 5\sqrt{2})\) using the difference of squares formula: \[ (x - 5\sqrt{2})(x + 5\sqrt{2}) = x^2 - (5\sqrt{2})^2 \]
Next, we simplify \((5\sqrt{2})^2\): \[ (5\sqrt{2})^2 = 25 \times 2 = 50 \]
Thus, the polynomial simplifies to: \[ x^2 - 50 \].So, our degree 2 polynomial that meets the given conditions is \( f(x) = x^2 - 50 \).

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