Problem 8 - 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 \(f(x)\) $$ \frac{1}{4}, \frac{3}{2} ; f(x)=8 x^{4}-30 x^{3}+23 x^{2}+8 x-3 $$

Step-by-Step Solution

Answer

Zeros are \(\frac{1}{4}, \frac{3}{2}, 1\). Factorization: \(f(x) = 8\left(x - \frac{1}{4}\right) \left(x - \frac{3}{2}\right) (x-1)^2\).

Step 1: Verify that \(\frac{1}{4}\) is a zero

Substitute \(x = \frac{1}{4}\) into the polynomial function \(f(x) = 8x^4 - 30x^3 + 23x^2 + 8x - 3\). Calculate the value to see if it equals zero.\[f\left(\frac{1}{4}\right) = 8\left(\frac{1}{4}\right)^4 - 30\left(\frac{1}{4}\right)^3 + 23\left(\frac{1}{4}\right)^2 + 8\left(\frac{1}{4}\right) - 3 = 0\]. Since the result is zero, \(\frac{1}{4}\) is a zero of the polynomial.

Step 2: Verify that \(\frac{3}{2}\) is a zero

Substitute \(x = \frac{3}{2}\) into the polynomial function \(f(x) = 8x^4 - 30x^3 + 23x^2 + 8x - 3\). Calculate the value to see if it equals zero. \[f\left(\frac{3}{2}\right) = 8\left(\frac{3}{2}\right)^4 - 30\left(\frac{3}{2}\right)^3 + 23\left(\frac{3}{2}\right)^2 + 8\left(\frac{3}{2}\right) - 3 = 0\]. Since the result is zero, \(\frac{3}{2}\) is a zero of the polynomial.

Step 3: Perform synthetic division for \(x - \frac{1}{4}\)

Using synthetic division, divide the polynomial \(f(x)\) by \(x - \frac{1}{4}\). The coefficient row is \([8, -30, 23, 8, -3]\). After applying synthetic division, the remainder is zero, confirming \(x - \frac{1}{4}\) is a factor. The quotient is \(8x^3 - 28x^2 + 16x + 12\).

Step 4: Perform synthetic division for \(x - \frac{3}{2}\)

Apply synthetic division on the quotient \(8x^3 - 28x^2 + 16x + 12\) by \(x - \frac{3}{2}\). The coefficient row for this is \([8, -28, 16, 12]\). After division, the remainder is zero, confirming \(x - \frac{3}{2}\) is also a factor. The quotient is \(8x^2 - 16x + 8\).

Step 5: Find remaining zeros

Factor the quotient \(8x^2 - 16x + 8\). Factor out 8 to get \(8(x^2 - 2x + 1)\). Recognize \(x^2 - 2x + 1 = (x-1)^2\), so \(x-1\) is a repeated zero.

Step 6: Write complete factorization

The complete factorization of \(f(x)\) with the known zeros is \[f(x) = 8\left(x - \frac{1}{4}\right) \left(x - \frac{3}{2}\right) (x-1)^2\].

Key Concepts

Synthetic Division

Synthetic division is a shortcut method to divide a polynomial by a binomial of the form \(x - c\). It’s faster and more efficient than traditional long division. The key advantage lies in its simplicity, especially when dealing with higher degree polynomials.
To perform synthetic division:

• Write down the coefficients of the polynomial in order of descending powers. For missing degrees, insert a 0 as a placeholder.
• Identify \(c\), the number being subtracted in the divisor \((x - c)\).
• Bring down the leading coefficient to the bottom row.
• Multiply \(c\) by the value just written on the bottom row and place it under the next coefficient.
• Add these two numbers and write the result in the bottom row.
• Repeat the previous two steps for all coefficients.

The numbers on the bottom row give you the coefficients of the quotient. If the last number is zero, it verifies that \(x - c\) is a factor of the polynomial. This was demonstrated twice in the original solution to confirm that \(x - \frac{1}{4}\) and \(x - \frac{3}{2}\) are factors.

Zeros of Polynomial

A zero of a polynomial is any value of \(x\) that makes the polynomial equal to zero. Knowing the zeros is crucial as they reveal where the graph of the polynomial intersects the x-axis.
In our original exercise, the function \(f(x) = 8x^4 - 30x^3 + 23x^2 + 8x - 3\) had its zeros initially verified by substituting the indicated numbers, \(\frac{1}{4}\) and \(\frac{3}{2}\), into the polynomial. When the polynomial evaluates to 0 at these values, it confirms they are zeros. Identifying zeros helps in determining the polynomial’s factorization, as each zero \(c\) corresponds to a factor \((x-c)\).
If a polynomial of degree \(n\) is known, it can have up to \(n\) zeros, including repeated ones. Zeros are essential to finding the polynomial’s factorization since the polynomial can be expressed as a product of its factors combined with any remaining quotient.

Polynomial Roots

Polynomial roots are essentially the same as the zeros of the polynomial. These are the solutions to the equation \(f(x) = 0\). Finding the roots is about solving for \(x\) when the polynomial is set to zero.
In the exercise, after identifying \(\frac{1}{4}\) and \(\frac{3}{2}\) as roots via synthetic division and substitution, the remaining polynomial \(8x^2 - 16x + 8\) was further factored. Factoring out the 8 and recognizing a perfect square \((x-1)^2\), we found a repeated root, \(x=1\).
Understanding the number and type of roots offers insights into the shape of the polynomial's graph. A root's multiplicity (how many times it’s repeated) affects the graph's behavior at that intercept. For example, a double root like \(x=1\) in this polynomial will have the graph tangentially touching the x-axis at \(x=1\). Knowing the roots enables complete polynomial factorization, facilitating easier graphing and interpretation of polynomial behavior.