Problem 48 - Algebra 2

Question

Given a function and one of its zeros, find all of the zeros of the function. \(f(x)=x^{3}+5 x^{2}+9 x+45 ;-5\)

Step-by-Step Solution

Answer

The zeros are \(-5\), \(3i\), and \(-3i\).

Step 1: Understanding the Problem

We are given the function \(f(x) = x^3 + 5x^2 + 9x + 45\) and one of its zeros, \(-5\). This means that \(f(-5) = 0\). Our goal is to find all the zeros of the function.

Step 2: Synthetic Division

To find other zeros, perform synthetic division using \(-5\) as the divisor. Arrange the coefficients of \(f(x)\) which are \(1, 5, 9, 45\). Start with the leading coefficient (1), multiply \(-5\) to it, add to the next coefficient, and repeat.

Step 3: Carry Out Synthetic Division

1. Bring down the 1. 2. Multiply \(1\) by \(-5\) to get \(-5\). Add to 5 to get 0. 3. Multiply 0 by \(-5\) to get 0. Add to 9 to get 9. 4. Multiply 9 by \(-5\) to get \(-45\). Add to 45 to get 0.The quotient polynomial is \(x^2 + 9\) and remainder is 0, confirming \(-5\) is indeed a zero.

Step 4: Finding Zeros of the Quotient Polynomial

Now evaluate \(x^2 + 9 = 0\). Solve for \(x\) by subtracting 9 from both sides and taking the square root:\[x^2 = -9\]\[x = \pm\sqrt{-9}\]This results in the complex zeros \(x = \pm 3i\).

Step 5: Conclusion of Zeros

Thus, the function \(f(x)\) has the zeros \(-5\), \(3i\), and \(-3i\).

Key Concepts

Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a binomial of the form \(x - c\). It simplifies the division process and helps to find zeroes quickly.
Instead of writing the whole polynomial, you use only the coefficients:

• Write the coefficients of the polynomial in a row.
• Write the zero, or root, you know outside the bracket.
• Bring down the first coefficient.
• Multiply this coefficient by the root and add the result to the next coefficient.
• Repeat this process for all coefficients.

The last number obtained is the remainder, and the others form the coefficients of the quotient polynomial. If the remainder is zero, the root is indeed a zero of the polynomial.
This process reduces the polynomial's degree by one, helping to further factor and find other zeros.

Complex Zeros

When finding zeros of a polynomial, you may encounter complex numbers. These zeros occur when taking square roots of negative numbers. Recall that the imaginary unit \(i\) is defined by \(i^2 = -1\).
For example, if you have an expression like \(x^2 + 9 = 0\), solving gives \(x = \pm\sqrt{-9}\), which results in \(x = \pm 3i\).
Complex zeros often come in conjugate pairs. This means if \(3i\) is a zero, then \(-3i\) is also a zero.

• They provide valuable insight into the polynomial's symmetry.
• Even if the polynomial's degree is odd, complex numbers could be part of its zeros.

Understanding complex zeros extends the reach of polynomial solutions beyond the real number line.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients.
They follow the format \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0\), where \(a_n\) are the coefficients and \(x\) is the variable.

• Higher degree polynomials have more complex structures and potentially more zeros.
• A polynomial of degree \(n\) can have up to \(n\) zeros, which might be real or complex.

Zeros of polynomial functions are the points where the polynomial evaluates to zero, commonly known as roots.
These zeros are significant in determining the function's graph behavior and intersections with the x-axis.
Learning to find and interpret these zeros helps in understanding the broader concepts of calculus and algebra.