Question

If \(f(x)=2 x^{3}+A x^{2}+4 x-5\) and \(f(2)=5,\) what is the value of \(A\) ?

Short answer

The value of A is -3.5.

Step-by-step solution

- Substitute x into the function

Start by substituting the given value of x (which is 2) into the function. The equation becomes: \[ f(2) = 2(2)^3 + A(2)^2 + 4(2) - 5 \]

- Simplify the equation

Simplify each term in the substituted equation: \[ 2(2)^3 = 2 \times 8 = 16 \] \[ A(2)^2 = A \times 4 = 4A \] \[ 4(2) = 8 \]\[ -5 = -5 \]Substitute these values back into the equation:\[ f(2) = 16 + 4A + 8 - 5 \]

- Set equation equal to given value

We know from the problem that \( f(2) = 5 \), so set the simplified equation equal to 5: \[ 16 + 4A + 8 - 5 = 5 \]

- Combine like terms

Combine the constant terms on the left-hand side of the equation: \[ 16 + 8 - 5 = 19 \] So the equation becomes: \[ 19 + 4A = 5 \]

- Solve for A

Isolate the term with A: \[ 4A + 19 = 5 \] Subtract 19 from both sides: \[ 4A = 5 - 19 \]Simplify: \[ 4A = -14 \]Finally, divide by 4 to solve for A: \[ A = \frac{-14}{4} = -3.5 \]

Key concepts

Solving Polynomial Equations

Understanding polynomial equations is critical. A polynomial equation is an expression of more than two algebraic terms, especially the sum or difference of several terms that contain different powers of the same variable(s). For example, in our exercise, the polynomial function is given as:
\[ f(x) = 2x^3 + Ax^2 + 4x - 5 \] Solving such an equation involves finding the value(s) of the variable(s) that make the equation true. Usually, these variables are denoted by letters like x in the given problem. Here, we have a specific case where we need to find the coefficient (A). To do that, we can use given values and the steps outlined above.

Substitution Method

The substitution method involves replacing variables with given values and simplifying the expression. This process helps in solving for unknown parameters. In our exercise, we were given that \[ f(2) = 5 \]. Here’s how substitution works:

1. Replace x with 2
2. Substitute it in the polynomial function: \[ f(2) = 2(2)^3 + A(2)^2 + 4(2) - 5 \] 3. Calculate each term:
- \[ 2(2)^3 = 16 \]
- \[ A(2)^2 = 4A \]
- \[ 4(2) = 8 \]
- \[ -5 = -5 \]
- Combine into: \[ 16 + 4A + 8 - 5 \]


Substitution helps in transforming complex equations into simpler forms, making it easier to solve for unknown variables.

Combining Like Terms

Combining like terms is essential in simplifying equations. This involves adding or subtracting the terms that have the same variable raised to the same power. In our case, after substituting and simplifying:
\[ 16 + 4A + 8 - 5 = 5 \]
We then combine the constants (16, 8, and -5) on the left-hand side:
\[ 16 + 8 - 5 = 19 \]
Thus, it transforms our equation to:
\[ 19 + 4A = 5 \]
Once combined, we solve for A by isolating it. Combining like terms reduces complexity and brings us closer to the solution. Here’s the step-by-step:

- Isolate \(A\): \[ 4A + 19 = 5 \]
- Subtract 19: \[ 4A = 5 - 19 \] - Simplify: \[ 4A = -14 \]
- Finally, divide by 4: \[ A = \frac{-14}{4} = -3.5 \]