Problem 13 One or more zeros are given for ... [FREE SOLUTION]

91影视

A Graphical Approach to Precalculus with Limits

John Hornsby, Margaret L. Lial, Gary Rockswold

$Math Studyset 91影视 Explanations$ Math

7 Edition

Chapter 4: Problem 13

One or more zeros are given for each polynomial. Find all remaining zeros. $P(x)=x^{3}-x^{2}-4 x-6 ; \quad 3$ is a zero.

Short Answer

Expert verified

The zeros of the polynomial are 3, $-1 + i$, and $-1 - i$.

Step by step solution

Verify the Given Zero

To confirm that 3 is a zero of the polynomial, substitute 3 into the polynomial equations. If it equals 0, then 3 is a zero.\[P(3) = 3^3 - 3^2 - 4(3) - 6 = 27 - 9 - 12 - 6 = 0\]Thus, 3 is indeed a zero of the polynomial.

Use Synthetic Division

Perform synthetic division of the polynomial by $x-3$, since 3 is a given zero. We write the coefficients of the polynomial and apply synthetic division:$\begin{array}{r|rrrr}3 & 1 & -1 & -4 & -6 \ & & 3 & 6 & 6 \\hline & 1 & 2 & 2 & 0 \\end{array}$The remainder is 0, and the quotient is $x^2 + 2x + 2$.

Find the Zeros of the Quotient

To find the remaining zeros of the polynomial, set the quotient equation $x^2 + 2x + 2 = 0$ equal to zero and solve for $x$ using the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]where $a = 1, b = 2, c = 2$.\[x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} = \frac{-2 \pm \sqrt{4 - 8}}{2} = \frac{-2 \pm \sqrt{-4}}{2} = \frac{-2 \pm 2i}{2}\]Simplifying gives:\[x = -1 \pm i\]

List All Zeros of the Polynomial

Combine the given zero with the zeros obtained from the quadratic formula:- The given zero of the polynomial is 3.- The zeros from the quotient are $-1 + i$ and $-1 - i$.Thus, the complete list of zeros of the polynomial is 3, $-1 + i$, and $-1 - i$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Synthetic Division

Synthetic division is a simplified way of dividing a polynomial by a linear factor of the form $x-c$, where $c$ is a constant. This method is particularly useful because it is less tedious and faster than traditional polynomial division.

To perform synthetic division:

Write down the coefficients of the polynomial. For example, in the polynomial $P(x)=x^{3}-x^{2}-4x-6$, the coefficients are 1, -1, -4, and -6.
Write the number corresponding to the linear factor. For $x-3$, this would be 3.
Start by bringing down the first coefficient (1 in this case) to the row below.
Multiply this number by the constant from the factor (3) and write it under the next coefficient.
Add the next coefficient and the result from the previous multiplication and write the result below the line.
Repeat the process with each coefficient.

The last number you get is the remainder, and if it's zero, $c$ is indeed a zero of the polynomial. The numbers in the bottom row (ignoring the remainder) give you the coefficients of the quotient polynomial.

Quadratic Formula

The quadratic formula is a powerful mathematical tool used to find the roots of any quadratic equation of the form $ax^2 + bx + c = 0$.

The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]To use it effectively:

Identify the coefficients $a$, $b$, and $c$ from your quadratic equation. Here, $a=1$, $b=2$, and $c=2$.
Substitute these values into the quadratic formula.
Calculate the discriminant, $b^2 - 4ac$. If it is positive, you'll find two real roots. If it's zero, there is one real root. If negative, the roots are complex numbers.
Complete the calculations to find the values of $x$.

In the exercise, the discriminant is $-4$, indicating that the roots are complex numbers, $-1 + i$ and $-1 - i$.

Complex Numbers

Complex numbers are numbers that include both a real component and an imaginary component. The imaginary unit is $i$, defined by the property $i^2 = -1$.

They are expressed in the form $a + bi$, where $a$ is the real part, and $b$ multiplied by $i$ is the imaginary part. For our polynomial quotient, we found the roots $-1 + i$ and $-1 - i$.

Understanding complex numbers:

They are essential in scenarios where equations have no real solutions (for example, when the discriminant is negative).
They help in describing periodic processes and waves, making them crucial in engineering and physics.
The arithmetic operations (addition, subtraction, multiplication, division) on complex numbers are often similar to real numbers, but pay attention to the $i^2$ rule.

Complex numbers allow mathematicians and scientists to work out solutions that aren't possible within the realm of real numbers alone.

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91影视

One or more zeros are given for each polynomial. Find all remaining zeros. \(P(x)=x^{3}-x^{2}-4 x-6 ; \quad 3\) is a zero.

Short Answer

Step by step solution

Verify the Given Zero

Use Synthetic Division

Find the Zeros of the Quotient

List All Zeros of the Polynomial

Key Concepts

Synthetic Division

Quadratic Formula

Complex Numbers

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