Question

In the form \(y=\) \(k \cdot(h(x))^{p}\) for some function \(h(x)\). $$ y=\left(1+x+x^{2}\right)^{3} $$

Short answer

Question: Identify the component parts of the function \(y = (1+x+x^2)^3\) and discuss the role each part plays in the function. Answer: The function \(y = (1+x+x^2)^3\) is in the form \(y= k \cdot(h(x))^p\), with: - Constant k: In this case, \(k=1\). This constant would affect the vertical stretch or compression of the function. - Function h(x): Here, \(h(x) = 1+x+x^2\). This is the inner function that is being transformed by the outer operation. - Constant p: The constant p is the power to which h(x) is raised, and in this case, \(p=3\). This power affects the shape and nature of the curve. In summary, the given function includes a constant k that affects vertical scaling (although k=1 doesn't change the function in this case), h(x) as the inner function being transformed, and a constant p that impacts the function's shape by raising h(x) to a power.

Step-by-step solution

Identify the form of the function

The given function is in the form \(y=(1+x+x^2)^3\). We can see that it matches the required form \(y= k \cdot(h(x))^p\).

Identify the constants and the function h(x)

In the given function, we can identify the following components: - Constant k: In our case, there is no constant directly multiplying the function, so we can say that \(k = 1\). - Function h(x): The function h(x) is located inside the parentheses. In this case, we have \(h(x) = 1+x+x^2\). - Constant p: The function h(x) is raised to the power of 3, therefore \(p = 3\).

Write the final form

We can rewrite the given function in the desired form, showing all constants and the function h(x): $$ y = 1 \cdot (1+x+x^2)^{3} $$ So the given function is in the form \(y= k \cdot(h(x))^p\), with \(k = 1\), \(h(x) = 1+x+x^2\), and \(p=3\).

Key concepts

Exponentiation

Exponentiation is a mathematical operation involving numbers, where a base is raised to the power of an exponent. For example, in the expression \[(1+x+x^2)^3\], the polynomial \((1+x+x^2)\) is the base, and 3 is the exponent. Exponentiation simplifies repeated multiplication of the same number.
Key properties include:

• Multiplying results in the power increasing: \((a^m) \cdot (a^n) = a^{m+n}\).
• Raising a power to another power: \((a^m)^n = a^{m \cdot n}\).
• For any number \(a\), \(a^0 = 1\) (as long as \(a\) is not zero).

These properties are widely used in algebraic operations, including simplifying expressions like in our example, rendering complex calculations more manageable. Understanding these concepts helps tackle various problems involving powers in mathematics.

Algebraic Form

The algebraic form of a polynomial can be thought of as an expression consisting of variables and constants combined using operations like addition and multiplication.
In our example, the polynomial \((1+x+x^2)\) represents an algebraic expression where each term \((1\), \(x\), and \(x^2)\) is connected through addition. The overall structure reflects the rules of algebra, allowing it to be manipulated systematically.
Important points of algebraic forms:

• Terms contain a coefficient (a constant multiplying the variable), like in \(2x\).
• Each term is separated by plus or minus signs.
• The degree of a polynomial is given by the highest power of the variable within the expression, which is 2 in \(1+x+x^2\).

Able to take on multiple forms, polynomials showcase how variables interact when combined using algebraic rules, crucial for progressing in more advanced topics.

Mathematical Constants

Mathematical constants are specific numbers with fixed values that appear in various equations and expressions.
In our example, the constant \(k\) is employed to scale the function and in this case, it is simply \(1\). Constants can arise in forms such as:

• Numerical values like \(\pi\) (approximated as 3.14159) and \(e\) (approximately 2.71828), both crucial in analytical and calculus contexts.
• Whole numbers symbolizing repeating units, like \(1\) and \(0\).

Constants play vital roles in mathematics, altering functions' effects without affecting their intrinsic formulations. Proper understanding of constants paves the way for clarity in algebraic manipulation and enhances comprehension of complicated expressions.