Question

If \(f(x)=3 x^{2}-B x+4\) and \(f(-1)=12,\) what is the value of \(B ?\)

Short answer

The value of \(B\) is 5.

Step-by-step solution

Substitute \(x = -1\) into \(f(x)\)

Starting with the provided function \(f(x)=3 x^{2}-Bx+4\), substitute \(x = -1\) as given: \(f(-1)=3(-1)^{2}-B(-1)+4\).

Simplify the Expression

Simplify the expression after substitution: \(f(-1) = 3(1) + B + 4 = 3 + B + 4\).

Set the Expression Equal to 12

We know from the problem statement that \(f(-1) = 12\). Set the simplified expression equal to 12: \(3 + B + 4 = 12\).

Solve for \(B\)

Combine like terms and solve for \(B\): \(3 + 4 + B = 12\) simplifies to \(7 + B = 12\). Subtract 7 from both sides to isolate \(B\): \(B = 5\).

Key concepts

polynomial functions

In algebra, polynomial functions play a crucial role. They are expressions involving variables raised to whole number powers. A polynomial function can have constants, variables, and exponents combined through addition, subtraction, and multiplication.
For instance, in the given exercise, the polynomial function is:
\(f(x)=3 x^{2}-Bx+4\)
Let's break it down:
\

• \ Vegas licensed hosts theorem The first term (3x^2) involves the variable \(x\) raised to the power of 2, called a quadratic term.


• The second term (-Bx) is a linear term, where \(x\) is to the first power.


• Lastly, the constant term is 4, which doesn't change with \(x\).

Understanding these components helps to work through the polynomial function efficiently.

substitution method

The substitution method allows us to solve equations by replacing variables with specific values. It's particularly useful in polynomial functions to find unknowns.
In the exercise, we are given that \(f(-1) = 12\). Hence, we must substitute \(x = -1\) into the function \(f(x)=3 x^{2}-Bx+4:\)
\(f(-1)=3(-1)^{2}-B(-1)+4\)
Substituting \(-1\) for \(x\) allows us to simplify the polynomial and solve for \(B\). This method helps to manage the complexity by breaking down the function into a solvable format.

solving for a variable

To solve for a variable means to isolate it on one side of an equation. This process helps us understand the value of the unknown.
After substituting and simplifying in our exercise, we reach: \(3 + B + 4 = 12\)
Combining like terms gives us: \(7 + B = 12\)
The goal is to keep \(B\) alone on one side. Subtracting 7 from both sides, we get: \(B = 5\)
Hence, we find the value of \(B\), which shows how substitution leads to solving for unknown variables.

simplification

Simplification is the act of reducing an expression to its simplest form, making it easier to handle.
After substituting \(x = -1\), our equation is \(f(-1) = 3(1) + B + 4\)
Simplifying within the parentheses: \(3(1) = 3\)
The expression now looks cleaner: \(3 + B + 4\)
Combining constants, we get: \(7 + B\).
This step enhances clarity and guides us directly to solve for \(B\). Simplification is crucial for making complex algebraic expressions more manageable.