Find A Degree 3 Polynomial With Real Coefficients Having Zeros 1 And $3i$, And A Leading Coefficient Of 1. Write $P(x$\] In Expanded Form. Be Sure To Write The Full Equation, Including $P(x)=$.

Understanding the Problem

When finding a polynomial with real coefficients, it's essential to remember that complex roots come in conjugate pairs. This means that if we have a complex root, its conjugate will also be a root of the polynomial. In this case, we're given a complex root of $3i$, so its conjugate, $-3i$, will also be a root.

Identifying the Roots

We have been given two roots: $1$ and $3i$. Since the coefficients are real, we know that the conjugate of $3i$, which is $-3i$, is also a root. Therefore, the three roots of the polynomial are $1$, $3i$, and $-3i$.

Constructing the Polynomial

To construct the polynomial, we can use the factored form of a polynomial, which is given by:

$$P(x) = a(x - r_1)(x - r_2)(x - r_3)$$

where $a$ is the leading coefficient, and $r_1$, $r_2$, and $r_3$ are the roots of the polynomial.

Finding the Factored Form

In this case, the leading coefficient is $1$, and the roots are $1$, $3i$, and $-3i$. Plugging these values into the formula, we get:

$$P(x) = 1(x - 1)(x - 3i)(x + 3i)$$

Simplifying the Factored Form

To simplify the factored form, we can multiply the factors together:

$$P(x) = (x - 1)(x^2 - (3i)^2)$$

Using the fact that $i^2 = -1$, we can simplify the expression further:

$$P(x) = (x - 1)(x^2 + 9)$$

Expanding the Polynomial

To expand the polynomial, we can multiply the factors together:

$$P(x) = x^2 + 9x - x - 9$$

Combining like terms, we get:

$$P(x) = x^2 + 8x - 9$$

The Final Answer

Therefore, the degree 3 polynomial with real coefficients having zeros $1$ and $3i$, and a leading coefficient of $1$ is:

$$P(x) = x^2 + 8x - 9$$

This polynomial has real coefficients and satisfies the given conditions.

Conclusion

In this article, we have found a degree 3 polynomial with real coefficients having zeros $1$ and $3i$, and a leading coefficient of $1$. We have used the factored form of a polynomial and simplified it to find the expanded form of the polynomial. The final answer is $P(x) = x^2 + 8x - 9$.

Q: What is the significance of complex roots coming in conjugate pairs?

A: Complex roots come in conjugate pairs because the coefficients of the polynomial are real. This means that if a complex number is a root of the polynomial, its conjugate will also be a root.

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

A: To find the conjugate of a complex number, you simply change the sign of the imaginary part. For example, the conjugate of $3i$ is $-3i$.

Q: What is the factored form of a polynomial?

A: The factored form of a polynomial is given by:

$$P(x) = a(x - r_1)(x - r_2)(x - r_3)$$

where $a$ is the leading coefficient, and $r_1$, $r_2$, and $r_3$ are the roots of the polynomial.

Q: How do I find the factored form of a polynomial?

A: To find the factored form of a polynomial, you need to know the roots of the polynomial. You can then plug these values into the formula for the factored form.

Q: What is the difference between the factored form and the expanded form of a polynomial?

A: The factored form of a polynomial is a product of factors, while the expanded form is a sum of terms. The factored form is often easier to work with, but the expanded form can be more useful for certain calculations.

Q: How do I expand a polynomial?

A: To expand a polynomial, you need to multiply the factors together. This can be done using the distributive property, which states that:

$$a(b + c) = ab + ac$$

Q: What is the distributive property?

A: The distributive property is a rule for multiplying expressions. It states that:

$$a(b + c) = ab + ac$$

This property can be used to expand polynomials and other algebraic expressions.

Q: How do I use the distributive property to expand a polynomial?

A: To use the distributive property to expand a polynomial, you need to multiply each term in one factor by each term in the other factor. This can be done using the formula:

$$a(b + c) = ab + ac$$

Q: What are some common mistakes to avoid when expanding a polynomial?

A: Some common mistakes to avoid when expanding a polynomial include:

• Forgetting to multiply each term in one factor by each term in the other factor
• Not using the distributive property correctly
• Not combining like terms

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms that have the same variable and exponent. For example:

$$2x + 3x = 5x$$

Q: What is the final answer to the problem of finding a degree 3 polynomial with real coefficients having zeros 1 and 3i, and a leading coefficient of 1?

A: The final answer is:

$$P(x) = x^2 + 8x - 9$$

This polynomial has real coefficients and satisfies the given conditions.