X^2 - 2x - 8: find the zeros of the quadratic polynomial and verify the relationship between the zeros and the coefficients.
Understand the Problem
The question is asking to determine the zeros of the quadratic polynomial X^2 - 2x - 8 and to check how these zeros relate to the coefficients of the polynomial according to algebraic principles.
Answer
The zeros are $X = 4$ and $X = -2$.
Answer for screen readers
The zeros of the polynomial $X^2 - 2X - 8$ are $X = 4$ and $X = -2$.
Steps to Solve
- Identify the polynomial and set it to zero
The polynomial given is $X^2 - 2X - 8$. To find the zeros, set the equation to zero:
$$ X^2 - 2X - 8 = 0 $$
- Factor the quadratic equation
Next, we will factor the quadratic equation. Look for two numbers that multiply to $-8$ (the constant term) and add to $-2$ (the coefficient of $X$). The numbers are $-4$ and $2$:
$$ (X - 4)(X + 2) = 0 $$
- Solve for X using the Zero Product Property
Using the Zero Product Property, we set each factor to zero:
$$ X - 4 = 0 \quad \text{or} \quad X + 2 = 0 $$
This gives us two equations to solve:
- From $X - 4 = 0$, we find $X = 4$.
- From $X + 2 = 0$, we find $X = -2$.
- List the zeros
The zeros of the polynomial $X^2 - 2X - 8$ are:
$$ X = 4 \quad \text{and} \quad X = -2 $$
- Relationship to the coefficients
According to Vieta's formulas, the sum of the roots (zeros) is equal to the opposite of the coefficient of $X$ (which is -2) and the product of the roots is equal to the constant term (-8):
- Sum: $4 + (-2) = 2$
- Product: $4 \cdot (-2) = -8$
This confirms that the zeros are related to the coefficients of the polynomial.
The zeros of the polynomial $X^2 - 2X - 8$ are $X = 4$ and $X = -2$.
More Information
Vieta's formulas indicate that for a quadratic polynomial of the form $ax^2 + bx + c$, the sum of the roots is $-\frac{b}{a}$ and the product is $\frac{c}{a}$. In this case, $b = -2$ and $c = -8$.
Tips
- Incorrect factoring: Ensure that the two numbers correctly multiply to the constant term and add up to the linear coefficient.
- Wrong roots: Double-check the equations set to zero to confirm the solutions for $X$.
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