91Ó°ÊÓ

In algebra, the zeros of a polynomial function are the values of x which make the polynomial equal to zero. These are sometimes referred to as the roots or solutions of the polynomial equation. For example, in the exercise, the given zeros are -5, \( \sqrt{3} \), and \( -\sqrt{3} \). This means that for these values of x, the function will be zero.
Understanding the concept of polynomial zeros is very important because it allows you to write the polynomial in a factored form. Each zero corresponds to a factor of the polynomial. If x = a is a zero, then \( (x - a) \) is a factor of the polynomial.
For the given zeros:
- Zero: -5 -> Factor: \( (x + 5) \)
- Zero: \( \sqrt{3} \) -> Factor: \( (x - \sqrt{3}) \)
- Zero: \( -\sqrt{3} \)-> Factor: \( (x + \sqrt{3}) \).

Conjugate pairs are pairs of numbers that differ only in the sign of their imaginary or irrational part. In the context of polynomials, these usually involve square roots or imaginary numbers. They are important because the product of conjugate pairs results in a rational number.
For example, the conjugate pairs given in the exercise are \( \sqrt{3} \) and \( -\sqrt{3} \). When these conjugate factors are multiplied:
\[ (x - \sqrt{3})(x + \sqrt{3}) \]
This simplifies to
\[ x^2 - (\sqrt{3})^2 = x^2 - 3 \]
Notice that the square root terms cancel out, leaving a polynomial with rational coefficients. This is a powerful technique for simplifying polynomial expressions.

Polynomial multiplication is the process of multiplying polynomials together to get a new polynomial. The process involves distributing each term of one polynomial to every term of the other polynomial.
In the exercise, after multiplying the conjugate factors, we obtained:
\[ (x - \sqrt{3})(x + \sqrt{3}) = x^2 - 3 \]
Next, we multiply this result by the remaining factor \( (x + 5) \):
\[ (x + 5)(x^2 - 3) \]
We use the distributive property to get:
\[ x(x^2 - 3) + 5(x^2 - 3) \]
This expands to:
\[ x^3 - 3x + 5x^2 - 15 \]
So, the polynomial after multiplication is:
\[ x^3 + 5x^2 - 3x - 15 \].

Polynomial expansion involves expressing a polynomial that is in a factored form into a standard form expression where the terms are no longer multiplied together. This usually results in a polynomial in the form \( ax^n + bx^{n-1} + ... + k \).
In the final step of the exercise, we expand \( (x + 5)(x^2 - 3) \).
We start with:
\[ (x + 5)(x^2 - 3) \]
Using the distributive property (also known as the FOIL method for binomials), we get:
\[ x(x^2 - 3) + 5(x^2 - 3) \]
Expanding inside each set of parentheses first, we obtain:
\[ x^3 - 3x + 5x^2 - 15 \]
Rewriting these terms in descending order of their degree, the fully expanded polynomial is:
\[ x^3 + 5x^2 - 3x - 15 \].
This final polynomial is the expanded form of the original factored expressions, and it is now ready to be used in various polynomial operations such as finding values, graphing, and more.