Problem 7 - Precalculus with Calculus Previews

Question

Verify that each of the indicated numbers are zeros of the given polynomial function \(f\). Find all other zeros and then give the complete factorization of \(f(x)\) $$ -3,5 ; f(x)=4 x^{4}-8 x^{3}-61 x^{2}+2 x+15 $$

Step-by-Step Solution

Answer

The zeros are \(-3, 5, \frac{1}{4}, -1\). Factorization is \(f(x) = (x + 3)(x - 5)(4x - 1)(x + 1)\).

Step 1: Evaluate the Polynomial at Given Zeros

First, substitute the indicated zeros \(-3\) and \(5\) into the function \(f(x)\) to verify that they are indeed zeros. This means we calculate \(f(-3)\) and \(f(5)\).Evaluate: - \(f(-3) = 4(-3)^4 - 8(-3)^3 - 61(-3)^2 + 2(-3) + 15 = 0\), which confirms that \(-3\) is a zero.- \(f(5) = 4(5)^4 - 8(5)^3 - 61(5)^2 + 2(5) + 15 = 0\), which confirms that \(5\) is also a zero.

Step 2: Use Synthetic Division to Divide Out Zeros

Now use synthetic division to divide the polynomial \(f(x)\) by \(x + 3\) and \(x - 5\), since we confirmed \(-3\) and \(5\) are zeros.- First, divide by \(x + 3\), resulting in a quotient polynomial. Repeat division of the result by \(x - 5\).After performing both divisions, the quotient should simplify to a lower degree polynomial: \(4x^2 + 3x - 1\).

Step 3: Solve for Remaining Zeros of the Quotient Polynomial

The quotient polynomial becomes \(4x^2 + 3x - 1\). Solve this quadratic equation using the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] where \(a=4\), \(b=3\), and \(c=-1\).Calculate:\[x = \frac{-3 \pm \sqrt{3^2 - 4(4)(-1)}}{2(4)} = \frac{-3 \pm \sqrt{9 + 16}}{8} = \frac{-3 \pm 5}{8}\]The solutions are \(x = \frac{1}{4}\) and \(x = -1\).

Step 4: State All Zeros and Factorize the Polynomial

List all zeros of the polynomial: \(-3, 5, \frac{1}{4}, -1\).Now, express the complete factorization as:\[f(x) = 4(x + 3)(x - 5)(x - \frac{1}{4})(x + 1)\].Rewriting the factors correctly with integer coefficients if needed:\[f(x) = (x + 3)(x - 5)(4x - 1)(x + 1)\].

Key Concepts

Zeros of Polynomials

The zeros of a polynomial are the values of the variable that make the polynomial equal to zero. Think of them as the points where the graph of the polynomial intersects the x-axis.
To find these zeros, you plug each value into the polynomial and determine if the result is zero.

• If the result is zero, that number is a zero of the polynomial.
• If not, it's not a zero.

In our exercise, we quickly checked whether

• -3 and 5 were zeros by substituting them into the polynomial.
• Since both evaluations resulted in zero, they are indeed zeros.

Recognizing zeros is crucial because they let you factor the polynomial into simpler parts.

Synthetic Division

Synthetic division is an efficient way to divide a polynomial by a binomial of the form

• \(x-c\), where \(c\) is a constant.

It simplifies the division process, allowing us to get a smaller degree polynomial quickly.
It is especially useful when you need to find remaining factors and zeros of the polynomial.

• First, we write out the coefficients of the polynomial.
• Then, we use the zero \(c\) and perform synthetic division.

In this exercise:

• We divided the polynomial by \(x+3\) and \(x-5\), using synthetic division because those were the given zeros.
• This allowed us to find that the remaining factor was a quadratic polynomial: \(4x^2 + 3x - 1\).

Quadratic Formula

The quadratic formula is a universal tool to solve any quadratic equation, specifically those in the form of

• \(ax^2 + bx + c\).

The formula is expressed as:

• \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

This magical formula allows us to find zeros of a quadratic without needing to factor by trial and error.
The term under the square root, \(b^2 - 4ac\), is called the discriminant and it tells us about the nature of the roots:

• If it's positive, we have two real and distinct roots.
• If it's zero, we have exactly one real root (a repeated root).
• If negative, we get complex roots.

For the polynomial \(4x^2 + 3x - 1\), we used this formula to find the remaining zeros: \(\frac{1}{4}\) and \(-1\).

Polynomial Factorization

Polynomial factorization involves breaking down a complex polynomial into simpler polynomials whose product gives back the original polynomial.
It's like taking apart the parts of a machine to understand each function better.

• This usually involves splitting the polynomial into factors based on its zeros.

Once we know all zeros, we can express the polynomial as a product of binomials.
In our exercise, factorization looked like this:

• We started with the polynomial \(f(x) = 4x^4 - 8x^3 - 61x^2 + 2x + 15\).
• We found its zeros: \(-3, 5, \frac{1}{4},\) and \(-1\).
• Then, we expressed it as a product of four linear expressions: \((x + 3)(x - 5)(4x - 1)(x + 1)\).

This process helps us understand the polynomial's behavior and simplifies further calculations.