Problem 43 Solve each equation. $$ 2 x^... [FREE SOLUTION]

Chapter 5: Problem 43

Solve each equation. $$ 2 x^{3}+x^{2}-98 x-49=0 $$

Short Answer

Expert verified

The solutions to the equation are \[ x = -\frac{1}{2}, -7, 7 \]

Step by step solution

- Identify the polynomial and its degree

The given equation is a polynomial equation: \[ 2x^3 + x^2 - 98x - 49 = 0 \]The degree of the polynomial is 3 since the highest power of the variable is 3.

- Factor by grouping

Group terms to find common factors: \[ 2x^3 + x^2 - 98x - 49 = (2x^3 + x^2) + (-98x - 49) \]Now factor each group: \[ x^2(2x + 1) - 49(2x + 1) \]

- Factor out the common binomial factor

Factor out the common binomial factor from both groups: \[ (2x + 1)(x^2 - 49) = 0 \]

- Solve for zero

Set each factor equal to zero and solve for x: 1. \[ 2x + 1 = 0 \]\[ x = -\frac{1}{2} \]2. \[ x^2 - 49 = 0 \]This is a difference of squares, which factors into: \[ (x + 7)(x - 7) = 0 \]So, \[ x + 7 = 0 \] leads to \[ x = -7 \]and\[ x - 7 = 0 \] leads to \[ x = 7 \]

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Key Concepts

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

Polynomial Degree

A polynomial's degree is the highest power of the variable in the equation.
For instance, in our exercise, the equation is \[ 2x^3 + x^2 - 98x - 49 = 0 \].
The highest power of $x$ here is 3 (from the term $2x^3$).

This means we are dealing with a cubic polynomial, as cubics are polynomials of degree 3.
Knowing the degree is crucial because it tells us how many solutions the equation can have.
For a cubic equation, there are up to 3 real solutions.

Factoring by Grouping

Factoring by grouping is a method to simplify polynomials by grouping terms that have common factors.
In our exercise, we start with the polynomial \[ 2x^3 + x^2 - 98x - 49 \].
First, we group the terms:
\[ (2x^3 + x^2) + (-98x - 49) \].
Next, we factor out the greatest common factor (GCF) from each group:
\[ x^2(2x + 1) - 49(2x + 1) \].
We see that $2x + 1$ is a common factor in both groups.
This allows us to factor it out:
\[ (2x + 1)(x^2 - 49) = 0 \].
This method helps in breaking down complex equations into simpler parts, making it easier to solve.

Difference of Squares

The difference of squares is a special factoring formula that looks like this:
\[ a^2 - b^2 = (a - b)(a + b) \].
In our problem, we have the term $x^2 - 49$.
Notice that 49 is a perfect square ($49 = 7^2$).
This means we can apply the difference of squares:
\[ x^2 - 49 = (x - 7)(x + 7) \].
This step is essential because it simplifies the polynomial further, helping us find the solutions.

Solving for Zero

To solve a polynomial equation like \[ 2x^3 + x^2 - 98x - 49 = 0 \], we need to find the values of $x$ that make the equation equal to zero.
After factoring, we get:
\[ (2x + 1)(x^2 - 49) = 0 \].
Set each factor to zero:
1. \[ 2x + 1 = 0 \], which gives us \[ x = -\frac{1}{2} \].
2. \[ x^2 - 49 = 0 \]. Using the difference of squares, we set \[ (x + 7) = 0 \] and \[ (x - 7) = 0 \], leading to solutions \[ x = -7 \] and \[ x = 7 \].
So, our solutions are $x = -\frac{1}{2}$, $x = -7$, and $x = 7$.
Solving for zero helps us find the roots of the polynomial, which are the values of $x$ that satisfy the equation.

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91影视

Solve each equation. $$ 2 x^{3}+x^{2}-98 x-49=0 $$

Short Answer

Step by step solution

- Identify the polynomial and its degree

- Factor by grouping

- Factor out the common binomial factor

- Solve for zero

Key Concepts

Polynomial Degree

Factoring by Grouping

Difference of Squares

Solving for Zero

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