Rationalizing the Coefficients

Oct 13, 2025 by windailys.com 140 views

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Introduction

In algebra, finding a polynomial function with rational coefficients that has specific zeros is a common problem. Given the zeros $5-i$ and $\sqrt{7}$ , we need to find a polynomial function of the lowest degree with rational coefficients that has these numbers as some of its zeros. In this article, we will explore the steps to find such a polynomial function.

Properties of Zeros

Before we start finding the polynomial function, let's recall some properties of zeros. If a polynomial function $f(x)$ has a zero at $a+bi$ , where $a$ and $b$ are real numbers and $i$ is the imaginary unit, then its complex conjugate $a-bi$ is also a zero of the polynomial function. This is known as the Complex Conjugate Root Theorem.

Step 1: Find the Complex Conjugate of the Given Zero

The given zero is $5-i$ . Using the Complex Conjugate Root Theorem, we can find its complex conjugate, which is $5+i$ .

Step 2: Find the Quadratic Factors

Since we have two zeros, $5-i$ and $5+i$ , we can write a quadratic factor that has these zeros. The quadratic factor is given by:

(x - (5 - i))(x - (5 + i))

Expanding this expression, we get:

(x - 5 + i)(x - 5 - i)

Using the difference of squares formula, we can simplify this expression to:

(x - 5)^2 - (i)^2

Simplifying further, we get:

x^2 - 10x + 25 - (-1)

x^2 - 10x + 26

This is the quadratic factor that has the zeros $5-i$ and $5+i$ .

Step 3: Find the Linear Factor

The given zero is $\sqrt{7}$ . Since this is a real number, we can write a linear factor that has this zero. The linear factor is given by:

(x - \sqrt{7})

Step 4: Find the Polynomial Function

Now that we have the quadratic factor and the linear factor, we can find the polynomial function by multiplying these factors together. The polynomial function is given by:

(x^2 - 10x + 26)(x - \sqrt{7})

Expanding this expression, we get:

x^3 - \sqrt{7}x^2 - 10x^2 + 10\sqrt{7}x + 26x - 26\sqrt{7}

Combining like terms, we get:

x^3 - (10 + \sqrt{7})x^2 + (10\sqrt{7} + 26)x - 26\sqrt{7}

This is the polynomial function in expanded form.

Conclusion

In this article, we found a polynomial function of the lowest degree with rational coefficients that has the given numbers as some of its zeros. The polynomial function is given by:

f(x) = x^3 - (10 + \sqrt{7})x^2 + (10\sqrt{7} + 26)x - 26\sqrt{7}

This polynomial function has the zeros $5-i$ , $5+i$ , and $\sqrt{7}$ .

Rationalizing the Coefficients

To rationalize the coefficients, we can multiply the polynomial function by the conjugate of the coefficient that contains the square root. In this case, we can multiply the polynomial function by $(\sqrt{7})^2 = 7$ .

Multiplying the polynomial function by 7, we get:

7x^3 - 7(10 + \sqrt{7})x^2 + 7(10\sqrt{7} + 26)x - 7(26\sqrt{7})

Expanding this expression, we get:

7x^3 - 70x^2 - 7\sqrt{7}x^2 + 70\sqrt{7}x + 182x - 182\sqrt{7}

Combining like terms, we get:

7x^3 - (70 + 7\sqrt{7})x^2 + (70\sqrt{7} + 182)x - 182\sqrt{7}

This is the polynomial function with rational coefficients.

Final Answer

The polynomial function in expanded form is:

f(x) = 7x^3 - (70 + 7\sqrt{7})x^2 + (70\sqrt{7} + 182)x - 182\sqrt{7}

This polynomial function has the zeros $5-i$ , $5+i$ , and $\sqrt{7}$ , and it has rational coefficients.

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Introduction

In our previous article, we found a polynomial function of the lowest degree with rational coefficients that has the given numbers as some of its zeros. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the Complex Conjugate Root Theorem?

A: The Complex Conjugate Root Theorem states that if a polynomial function $f(x)$ has a zero at $a+bi$ , where $a$ and $b$ are real numbers and $i$ is the imaginary unit, then its complex conjugate $a-bi$ is also a zero of the polynomial function.

Q: How do I find the complex conjugate of a given zero?

A: To find the complex conjugate of a given zero, you can simply change the sign of the imaginary part. For example, if the given zero is $5-i$ , the complex conjugate is $5+i$ .

Q: What is the quadratic factor that has the zeros $5-i$ and $5+i$ ?

A: The quadratic factor that has the zeros $5-i$ and $5+i$ is given by:

(x - (5 - i))(x - (5 + i))

Expanding this expression, we get:

(x - 5 + i)(x - 5 - i)

Using the difference of squares formula, we can simplify this expression to:

(x - 5)^2 - (i)^2

Simplifying further, we get:

x^2 - 10x + 25 - (-1)

x^2 - 10x + 26

Q: How do I find the linear factor that has the zero $\sqrt{7}$ ?

A: The linear factor that has the zero $\sqrt{7}$ is given by:

(x - \sqrt{7})

Q: How do I find the polynomial function by multiplying the quadratic factor and the linear factor?

A: To find the polynomial function, you can multiply the quadratic factor and the linear factor together. The polynomial function is given by:

(x^2 - 10x + 26)(x - \sqrt{7})

Expanding this expression, we get:

x^3 - \sqrt{7}x^2 - 10x^2 + 10\sqrt{7}x + 26x - 26\sqrt{7}

Combining like terms, we get:

x^3 - (10 + \sqrt{7})x^2 + (10\sqrt{7} + 26)x - 26\sqrt{7}

Q: How do I rationalize the coefficients of the polynomial function?

A: To rationalize the coefficients, you can multiply the polynomial function by the conjugate of the coefficient that contains the square root. In this case, you can multiply the polynomial function by $(\sqrt{7})^2 = 7$ .

Multiplying the polynomial function by 7, we get:

7x^3 - 7(10 + \sqrt{7})x^2 + 7(10\sqrt{7} + 26)x - 7(26\sqrt{7})

Expanding this expression, we get:

7x^3 - 70x^2 - 7\sqrt{7}x^2 + 70\sqrt{7}x + 182x - 182\sqrt{7}

Combining like terms, we get:

7x^3 - (70 + 7\sqrt{7})x^2 + (70\sqrt{7} + 182)x - 182\sqrt{7}

Q: What is the final answer to the problem?

A: The polynomial function in expanded form is:

f(x) = 7x^3 - (70 + 7\sqrt{7})x^2 + (70\sqrt{7} + 182)x - 182\sqrt{7}

This polynomial function has the zeros $5-i$ , $5+i$ , and $\sqrt{7}$ , and it has rational coefficients.

Conclusion

In this article, we answered some frequently asked questions related to finding a polynomial function of the lowest degree with rational coefficients that has the given numbers as some of its zeros. We hope that this article has been helpful in understanding this topic.