Problem 10 - College Algebra and Calculus: An Applied Approach

Question

Determine whether each value of \(x\) is a solution of the equation. Equation $$ 5 x^{3}+2 x-3=4 x^{3}+2 x-11 $$ Values (a) \(x=2\) (b) \(x=-2\) (c) \(x=0\) (d) \(x=10\)

Step-by-Step Solution

Answer

Among the provided values, only \(x = -2\) is a valid solution to the equation. The other values (\(x = 2\), \(x = 0\), and \(x = 10\)) are not solutions.

Step 1: Solution Test for value \(x = 2\)

Substitute \(x = 2\) to the equation:\(5(2)^{3}+2(2)-3 = 4(2)^{3}+2(2)-11\)From this we can calculate:\(40 + 4 - 3 = 32 + 4 - 11\)That simplifies to:\(41 = 25\)So, \(x = 2\) is not a solution since the equality doesn't hold.

Step 2: Solution Test for value \(x = -2\)

Substitute \(x = -2\) to the equation:\(5(-2)^{3}+2(-2)-3 = 4(-2)^{3}+2(-2)-11\)From this we can calculate:\(-40 - 4 - 3 = -32 - 4 - 11\)That simplifies to:\(-47 = -47\)So, \(x = -2\) is a solution since the equality holds.

Step 3: Solution Test for value \(x = 0\)

Substitute \(x = 0\) to the equation:\(5(0)^{3}+2(0)-3 = 4(0)^{3}+2(0)-11\)From this we can calculate:\(-3 = -11\)So, \(x = 0\) is not a solution since the equality doesn't hold.

Step 4: Solution Test for value \(x =10\)

Substitute \(x = 10\) to the equation:\(5(10)^{3}+2(10)-3 = 4(10)^{3}+2(10)-11\)From this we can calculate:\(5000 + 20 -3 = 4000 + 20 -11 \)That simplifies to:\(5017 = 4009 \)So, \(x = 10\) is not a solution since the equality doesn't hold.

Key Concepts

Polynomial Equality

Understanding polynomial equality is essential for solving polynomial equations. In the given exercise, we see a cubic polynomial on both sides of the equation, namely 5x^3 + 2x - 3 and 4x^3 + 2x - 11. Polynomial equality implies that the two expressions are identical for all values of x.

However, most polynomials are not equal across all values of x, and hence we may need to find specific instances where equality holds. By equating the two expressions, we begin the journey to discovering potential solutions where the polynomials evaluate to the same number. This is foundational in algebra, where we often seek to solve for unknown variables by setting two expressions equal to each other and simplifying the equation.

Cubic Equations

Cubic equations, like the one in this exercise, are polynomial equations where the highest exponent of the variable x is 3. They generally take the form of ax^3 + bx^2 + cx + d = 0, with a, b, c, and d representing constants, and a ≠ 0 to ensure the equation is indeed cubic.

Solving cubic equations can involve various techniques, such as factoring by grouping, synthetic division, or applying the Rational Root Theorem. However, for verifying whether a particular value is a solution, substitution—as demonstrated in the step-by-step solution—is usually the quickest method. By substituting the given values into the equation, we effectively evaluate whether the two sides of the equation balance, which is a clear indication that we have found a solution.

Solution Verification

Solution verification is a critical final step in the problem-solving process. After we perform operations such as substitution, as shown in the steps for the given exercise, we must compare the results to determine if the value is a true solution.

When the left side of the equation matches the right side after substitution, as seen with x = -2, we confirm that the given value is indeed a solution. If they do not match, as with x = 2, x = 0, and x = 10, the value is not a solution. This verification process is vital because it helps us identify mistakes and ensures that our solutions are valid. It's also important to note that solution verification applies not only to cubic equations but to any algebraic equation where solutions can be proposed.