Polynomial Functions - Polynomial Function or NOT? Grade 10 Math Second Quarter

Summary

In this educational video, the teacher explains the fundamentals of polynomial functions, including their general form and key characteristics. The video provides examples to distinguish between polynomial and non-polynomial functions, highlighting common mistakes like having variables under radicals or fractional exponents. Additionally, the teacher demonstrates how to determine the degree, leading coefficient, and constant term of a polynomial. By the end, students gain a clear understanding of how to identify and analyze polynomial functions with real-life examples.

Takeaways

• 😀 A polynomial function is generally expressed as P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where each term has a coefficient and a non-negative integer exponent.
• 😀 Polynomial functions involve variables raised to whole number powers only; no radicals, fractions, or negative exponents are allowed.
• 😀 Examples like y = 14x and y = -2022x are valid polynomial functions.
• 😀 Expressions such as y = 5x^3 - 4√(2x) or y = x^(3/4) + 3x^(1/4) + 7 are not polynomials due to radicals or fractional exponents.
• 😀 Variables in the denominator (e.g., y = 1/(2x^3) + 2/(3x^4) - 3/(4x^5)) make a function non-polynomial because they produce negative exponents.
• 😀 The degree of a polynomial is the highest exponent of the variable in the function.
• 😀 The leading coefficient is the coefficient of the term with the highest degree.
• 😀 The constant term is the term without any variable.
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• 😀 Simplifying expressions is essential to correctly identify degree, leading coefficient, and constant term (e.g., y = x(x^2 - 5) becomes y = x^3 - 5x).
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• 😀 Properly identifying polynomial functions involves checking for restrictions: no radicals, no fractional exponents, and no negative exponents.

Q & A

What is the general form of a polynomial function?

A polynomial function can be represented as P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_2 x^2 + a_1 x + a_0, where a_n, a_{n-1}, ..., a_0 are constants, x is the variable, and n is a non-negative integer.

What are the restrictions for a function to be considered a polynomial?

A function is a polynomial if it does not have variables inside radical signs, fractional exponents, or negative exponents (no variables in denominators).

Is y = 14x a polynomial function and why?

Yes, y = 14x is a polynomial function because it has a variable with a non-negative integer exponent and no restrictions are violated.

Why is y = 5x^3 - 4√(2x) + x not a polynomial function?

It is not a polynomial function because the variable x is inside a radical (square root), which is not allowed in polynomial functions.

Why is y = x^(3/4) + 3x^(1/4) + 7 not a polynomial function?

It is not a polynomial function because it contains fractional exponents, which are not permitted in polynomials.

How can y = 1/2 x^3 + 2/3 x^4 - 3/4 x^5 be rewritten, and why is it not a polynomial?

It can be rewritten as (1/2)x^3 + (2/3)x^4 - (3/4)x^5, but if expressed with the variable in the denominator it would have negative exponents, which violates polynomial rules.

What is the degree, leading coefficient, and constant term of f(x) = 2x^2 - 11x + 2?

The degree is 2 (highest exponent), the leading coefficient is 2 (coefficient of the leading term), and the constant term is 2.

How do you determine the degree of a polynomial function?

The degree of a polynomial function is determined by the highest exponent of the variable after simplifying the function.

What is the degree, leading coefficient, and constant term of y = x(x^2 - 5)?

After simplifying, y = x^3 - 5x. The degree is 3, the leading coefficient is 1, and the constant term is 0.

What is the degree, leading coefficient, and constant term of y = x^4 + 2x^3 - x^2 + 14x - 56?

The degree is 4, the leading coefficient is 1, and the constant term is -56.

Why is y = -2022x considered a polynomial function?

It is considered a polynomial function because it has a single term with a variable raised to a non-negative integer exponent and does not violate any polynomial restrictions.