In Exercises 91–100, find all values of x satisfying the given conditions. $y=x^3+4x^2-x+6$ and $y=10$

Verified step by step guidance

1. Start with the given equations: \( y = x^3 + 4x^2 - x + 6 \) and \( y = 10 \). Since both expressions equal \( y \), set them equal to each other: \( x^3 + 4x^2 - x + 6 = 10 \).
2. Subtract 10 from both sides to set the equation to zero: \( x^3 + 4x^2 - x + 6 - 10 = 0 \), which simplifies to \( x^3 + 4x^2 - x - 4 = 0 \).
3. Now, you have a cubic equation \( x^3 + 4x^2 - x - 4 = 0 \). The next step is to try to find rational roots using the Rational Root Theorem, which suggests possible roots are factors of the constant term divided by factors of the leading coefficient.
4. Test possible rational roots (such as \( \pm1, \pm2, \pm4 \)) by substituting them into the cubic equation to see if they satisfy it (i.e., make the equation equal to zero).
5. Once a root is found, use polynomial division or synthetic division to factor the cubic polynomial into a linear factor and a quadratic factor. Then solve the quadratic factor using the quadratic formula or factoring to find the remaining roots.

Key Concepts

Solving Polynomial Equations

Solving polynomial equations involves finding the values of the variable that make the polynomial equal to a given number. In this case, setting y = 10 means solving the cubic equation x^3 + 4x^2 - x + 6 = 10. This requires rearranging the equation and finding roots that satisfy it.

Setting Equations Equal to Each Other

When two expressions for y are given, finding x values that satisfy both means setting the expressions equal or substituting the given y value. Here, since y = 10, we substitute 10 into the polynomial and solve for x, turning the problem into finding roots of a single-variable equation.

Methods for Finding Roots of Cubic Equations

Cubic equations can be solved using factoring, synthetic division, or numerical methods like the Rational Root Theorem or graphing. Identifying possible rational roots and testing them helps simplify the cubic to find all solutions for x that satisfy the equation.