Question

Evaluate the function for the given value of \(x\). $$h(x)=-x^3+3 x^2+5 x ; x=3$$

Short answer

The evaluated value of the function for \(x=3\) is \(15\).

Step-by-step solution

Substitute the given value of \(x\) into the function

First, the value of \(x\) which is \(3\) needs to be substituted into the function \(h(x)=-x^3+3x^2+5x\). So, \(h(3)=-3^3+3*3^2+5*3\).

Follow the order of operations

Now, follow the order of operations specified by BIDMAS or PEMDAS which means first calculate the values of exponents. So, \(h(3)=-27+3*9+15 = -27+27+15\).

Perform the addition/subtraction operation

Finally, perform the addition/subtraction operations, which concludes that \(h(3)=15\).

Key concepts

Polynomial Functions

Polynomial functions are mathematical expressions that consist of variables raised to whole number exponents and multiplied by constants, called coefficients. They can take various forms, from simple linear equations to more complex cubic or quartic equations. In this exercise, we dealt with a cubic polynomial function, which means the highest exponent of the variable is three. Polynomial functions can be represented as:

• The sum of terms: each term consists of a coefficient, a variable, and an exponent.
• In the given function, \(h(x) = -x^3 + 3x^2 + 5x\), we see three terms where \(3\) is the highest power of \(x\).
• The leading term, \(-x^3\), defines the degree and basic shape of the polynomial.

Understanding polynomial functions is crucial as they model many real-world situations, such as calculating trajectories, profit margins, and trends. They are diverse and adaptable, making them a cornerstone in mathematics.

Order of Operations

The order of operations is a fundamental concept in mathematics that ensures consistent results when evaluating expressions. The commonly remembered acronym BIDMAS or PEMDAS helps guide us:

• B - Brackets first.
• I/D - Indices (exponents) or Division.
• M - Multiplication.
• A/S - Addition and Subtraction.

In this problem, after substituting \(x=3\) into the polynomial \(h(x) = -x^3 + 3x^2 + 5x\), we applied the order of operations:

• First, calculate the exponents: \(-3^3\) and \(3^2\).
• Second, carry out any multiplication involved: \(3 \cdot 9\) and \(5 \cdot 3\).
• Finally, perform addition and subtraction from left to right: \(-27 + 27 + 15\).

The accuracy of our calculations hinges on following this order, leading us to the correct result of \(h(3) = 15\). Skipping or rearranging steps can lead to incorrect answers.

Substitution Method

The substitution method is a technique used to simplify and solve mathematical problems. This method involves replacing variables in an expression with given numerical values to evaluate the expression.For the function \(h(x) = -x^3 + 3x^2 + 5x\), we replaced \(x\) with its specific value, \(3\), directly into the equation.This step-by-step approach makes complex equations more manageable by reducing them into simpler arithmetic operations:

• You substitute the given value directly from the problem into the expression.
• It allows analysis of specific cases of a more general expression.
• In this case, \(h(3)\) was evaluated by substituting \(3\) for \(x\), resulting in the expression \(-3^3 + 3 \cdot 3^2 + 5 \cdot 3\).

Using substitution correctly helps obtain precise calculations and makes it easier to interpret results within the context of the original equation.