Problem 14 Find the remaining roots of the ... [FREE SOLUTION]

Chapter 15: Problem 14

Find the remaining roots of the given equations using synthetic division, given the roots indicated. $$x^{4}-2 x^{3}-20 x^{2}-8 x-96=0 \quad\left(r_{1}=6, r_{2}=-4\right)$$

Short Answer

Expert verified

The remaining roots are $+2i$ and $-2i$.

Step by step solution

Verify Known Roots

First, plug the known roots $ r_1 = 6 $ and $ r_2 = -4 $ into the polynomial equation to verify they are indeed roots. This step confirms that the given solutions are correct.For $ r_1 = 6 $:\[ f(6) = 6^4 - 2(6)^3 - 20(6)^2 - 8(6) - 96 = 0 \] For $ r_2 = -4 $:\[ f(-4) = (-4)^4 - 2(-4)^3 - 20(-4)^2 - 8(-4) - 96 = 0 \] Both calculations confirm that these roots satisfy the equation.

Perform Synthetic Division with $ r_1 = 6 $

Use synthetic division to divide the polynomial by $ x - 6 $.Write down the coefficients: 1, -2, -20, -8, -96.* Drop the first coefficient (1).* Multiply by 6 (root) to get 6; add this to the second coefficient (-2): $-2 + 6 = 4$.* Multiply 4 by 6 to get 24; add this to the next coefficient (-20): $-20 + 24 = 4$.* Multiply 4 by 6 again to get 24; add this to -8: $-8 + 24 = 16$.* Multiply 16 by 6 for 96, add this to -96: $-96 + 96 = 0$.The quotient polynomial is $x^3 + 4x^2 + 4x + 16$.

Perform Synthetic Division with $ r_2 = -4 $

Now, divide the quotient polynomial $ x^3 + 4x^2 + 4x + 16 $ by $ x + 4 $ using synthetic division.Coefficients to use: 1, 4, 4, 16.* Drop the first coefficient (1).* Multiply by -4 to get -4; add to 4: $4 - 4 = 0$.* Multiply 0 by -4 to get 0; add to 4: $4 + 0 = 4$.* Multiply 4 by -4 to get -16; add to 16: $16 - 16 = 0$.The resulting polynomial is $x^2 + 4$.

Find Remaining Roots

The simplified polynomial is $x^2 + 4$. Set this equal to zero and solve for $x$:\[ x^2 + 4 = 0 \] Subtract 4 from both sides:\[ x^2 = -4 \]Take the square root of both sides:\[ x = \pm \sqrt{-4} = \pm 2i \]Thus, the remaining roots are $+2i$ and $-2i$.

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Key Concepts

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

Polynomial Roots

Finding the roots of a polynomial is like unraveling a mystery. These roots are the values where the polynomial equals zero. They are the 'solutions' to the polynomial equation:

If given a polynomial $ f(x) = 0 $, the roots are the $ x $-values that make the equation true.
Think of roots as the points on a graph where the polynomial curve touches or crosses the x-axis.
For example, in $ x^4 - 2x^3 - 20x^2 - 8x - 96 = 0 $, roots help us understand where the graph intersects zero.

Knowing some of the roots, like 6 and -4 in this problem, can simplify solving the polynomial. With synthetic division, we leverage known roots to break down large problems. This way, we focus on finding simpler, smaller-degree polynomials.

Complex Numbers

In mathematics, not all polynomial roots are 'real'. Sometimes, especially when dealing with quadratic terms or negative numbers under a square root, we encounter complex numbers:

A complex number is expressed as $ a + bi $.
Here, $ a $ is the real part, and $ bi $ is the imaginary part.
The imaginary unit $ i $ is defined such that $ i^2 = -1 $.

For instance, in the equation $ x^2 + 4 = 0 $, solving gives $ x = \pm 2i $. These solutions include imaginary numbers, highlighting that not all solutions are visible on the real number line. Acknowledging complex solutions is vital for a complete understanding of polynomial roots.

Verification of Roots

Verification ensures that assumed roots are indeed solutions. It's crucial when using methods like synthetic division, which rely on accurate input:

Plug a root into the original polynomial equation.
If substituting gives zero, it's a true root.
This ensures that division will simplify the polynomial correctly.

In the given exercise, computing $ f(6) $ and $ f(-4) $ and confirming both equal zero verified the correctness of the initial roots. This step cannot be overlooked, as it lays the foundation for breaking down the polynomial into simpler forms and uncovering other possible roots systematically and accurately.

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91影视

Find the remaining roots of the given equations using synthetic division, given the roots indicated. $$x^{4}-2 x^{3}-20 x^{2}-8 x-96=0 \quad\left(r_{1}=6, r_{2}=-4\right)$$

Short Answer

Step by step solution

Verify Known Roots

Perform Synthetic Division with \( r_1 = 6 \)

Perform Synthetic Division with \( r_2 = -4 \)

Find Remaining Roots

Key Concepts

Polynomial Roots

Complex Numbers

Verification of Roots

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