Problem 18 Determine which functions are po... [FREE SOLUTION] (2024)

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The degree of a polynomial is an essential concept to understand. It's the highest power (exponent) of the variable present in the polynomial. For example, in the polynomial function \( h(x) = -\frac{1}{2} x + 3 \), the highest exponent is 1, so the degree of this polynomial is 1.
The degree helps categorize polynomial functions and gives insight into their behavior and graph shapes.
It's important to remember that:

• Constant terms have a degree of 0.
• A polynomial like \( x^3 + 2x^2 + x + 1 \) has a degree of 3 because the term \( x^3 \) has the highest exponent.

If you encounter terms with variables in a potential polynomial function, always look for the term with the largest exponent to determine its degree.

The leading term of a polynomial is the term with the highest power of the variable. It is significant not only because it tells us about the degree of the polynomial, but it also has the most influence on the polynomial's behavior for large values of the variable.
For the polynomial \( h(x) = -\frac{1}{2} x + 3 \), the leading term is \( -\frac{1}{2} x \).
The leading term's coefficient and exponent affect the end behavior of polynomial functions. For instance:

• In \( 3x^2 + 2x + 1 \), the leading term is \( 3x^2 \).
• For \( x^4 - 4x^3 + x^2 \), the leading term is \( x^4 \).

This term often dictates the shape and direction of the polynomial graph as you move away from the origin.

In a polynomial, the constant term is the term that has no variable attached to it. It's a standalone number that stays the same regardless of the value of the variable.
In our example polynomial \( h(x) = -\frac{1}{2} x + 3 \), the constant term is 3.
It's important to identify the constant term because:

• It tells you the y-intercept of the polynomial when graphed.
• It helps in understanding the polynomial's overall value at zero (\( x = 0 \)).

For example:

• In \( 4x^2 + 2x + 5 \), the constant term is 5.
• For \( x^3 - 4x + 7 \), the constant term is 7.
• In polynomials like \( -6x^4 + 2x^3 \), while there is no explicit constant term written, it is understood to be 0.

Writing a polynomial in its standard form means arranging the terms in descending order of their exponents, from the highest to the lowest. This makes it easier to identify the degree, leading term, and the polynomial's overall structure.
For the function \( h(x) = 3 - \frac{1}{2} x \), the standard form is \( h(x) = -\frac{1}{2} x + 3 \). Here's how you do it:

• Ensure all terms are ordered by decreasing exponent value.
• Include all terms, even the constant term.

For example:

• \( 5 - x + x^2 \) in standard form is \( x^2 - x + 5 \).
• \( 2x - 7 + 3x^3 \) in standard form is \( 3x^3 + 2x - 7 \).

Writing polynomials in standard form simplifies operations like addition, subtraction, and factoring.