Problem 67 - Algebra 2

Question

Find \(p(7)\) and \(p(-3)\) for each function. $$ p(x)=\frac{2}{3} x^{4}-3 x^{3} $$

Step-by-Step Solution

Answer

\(p(7) = 571.67\) and \(p(-3) = 135\).

Step 1: Substitute 7 into the function

To find \(p(7)\), substitute \(x = 7\) into the function: \[p(7) = \frac{2}{3} \times 7^4 - 3 \times 7^3\] Calculate each term separately to simplify.

Step 2: Calculate powers of 7

First, calculate \(7^4\) and \(7^3\). \[7^4 = 2401\] \[7^3 = 343\]

Step 3: Evaluate the expression for x = 7

Substitute the calculated powers into the expression: \[p(7) = \frac{2}{3} \times 2401 - 3 \times 343\] This simplifies to: \[p(7) = \frac{4802}{3} - 1029\] Solve this to find \(p(7)\).

Step 4: Simplify the calculation for p(7)

Calculate \(\frac{4802}{3} = 1600.67\). Subtract 1029 from this value: \[p(7) = 1600.67 - 1029 = 571.67\]

Step 5: Substitute -3 into the function

To find \(p(-3)\), substitute \(x = -3\) into the function: \[p(-3) = \frac{2}{3} \times (-3)^4 - 3 \times (-3)^3\] Calculate each term separately to simplify.

Step 6: Calculate powers of -3

First, calculate \((-3)^4\) and \((-3)^3\). \[(-3)^4 = 81\] \[(-3)^3 = -27\]

Step 7: Evaluate the expression for x = -3

Substitute the calculated powers into the expression: \[p(-3) = \frac{2}{3} \times 81 - 3 \times (-27)\] This simplifies to: \[p(-3) = \frac{162}{3} + 81\] Solve this to find \(p(-3)\).

Step 8: Simplify the calculation for p(-3)

Calculate \(\frac{162}{3} = 54\). Add 81 to this value: \[p(-3) = 54 + 81 = 135\]

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Here's the breakdown of what makes a polynomial function:

• It involves variables such as \(x\), which can be raised to whole number exponents.
• Coefficients are the numbers in front of the variables (e.g., \(\frac{2}{3}\) and \(-3\) in our example).
• The degree of the polynomial is determined by the highest power of the variable in the expression. For instance, \(x^4\) has the highest power, so the degree here is 4.

In our example, \(p(x) = \frac{2}{3}x^4 - 3x^3\), we see a polynomial of degree 4 because the highest power is 4. Each component of the polynomial functions can be distinctly evaluated, and this is key when performing operations or substituting values.

Substitution Method

The substitution method is a foundational technique in evaluating polynomial functions. It involves replacing the variable in the polynomial equation with a specific numerical value. This is how it works:

• Identify the polynomial function and choose the value to substitute.
• Replace every instance of the variable with the chosen number.
• Follow through with calculations, respecting the order of operations (PEMDAS/BODMAS).

In our example, we substitute \(x = 7\) and \(x = -3\) to evaluate \(p(7)\) and \(p(-3)\), respectively. This helps in determining the value of the function at particular points, which can reveal insights about the function's behavior. Substitution is an essential procedure in mathematics for evaluating, verifying solutions, and solving equations.

Exponentiation

Exponentiation is a crucial operation when working with polynomial functions. It involves raising numbers to specific powers to express repeated multiplication. Here are fundamental points:

• An exponent indicates how many times a number, known as the base, is multiplied by itself.
• For instance, \(7^4\) means multiplying 7 by itself four times: \(7 \times 7 \times 7 \times 7\).
• Negative bases follow different rules; for example, \((-3)^3\) means \((-3) \times (-3) \times (-3)\), resulting in -27.

In the given example, this allows us to evaluate terms like \(7^4\) and \(7^3\) when calculating \(p(7)\), and similarly \((-3)^4\) and \((-3)^3\) for \(p(-3)\). Understanding exponentiation is fundamental to solving many polynomial equations effectively.

Simplification of Expressions

The simplification of expressions is the process of clarifying and reducing complex mathematical expressions to a simpler form without changing its value. This involves:

• Combining like terms, where possible.
• Performing operations such as addition, subtraction, and division as necessary.
• Ensuring fractions are simplified to their lowest term, if involved.

In our example, after substituting \(x\) values and calculating powers, we simplify the expression by performing operations like multiplication and subtraction. Please see, for \(p(7)\), it involves calculating \(\frac{4802}{3} - 1029\), which simplifies to 571.67. Similarly, for \(p(-3)\), it simplifies to 135 after performing the addition and simplification step. This simplification makes expressions easier to understand and work with, highlighting the result of our valuations clearly.