Polynomial Evaluation and Zeroes Verification Quiz

For the polynomial y2 – 5y + 6, which of the following accurately describes the factors found using the Factor Theorem?

What method is deemed more efficient in the text for factorizing polynomials like y2 – 5y + 6?

Based on the text, how are factors of a polynomial related to its constant term?

In Example 10, what is the correct factorization of x3 – 23x2 + 142x – 120?

What is the value of p(2) for the polynomial y2 – 5y + 6 in the text?

Why does the text suggest starting with finding at least one factor when factorizing cubic polynomials?

What role does a play in factorizing polynomials according to the Factor Theorem?

Which method is not suitable to start with for factorizing cubic polynomials based on the information provided?

'ab = 6' indicates that in p(y), a and b are _________.

'p(x) = (x – a) (x – b)' represents what form of polynomial factorization according to the text?

Polynomials Overview

A polynomial consists of multiple terms; for example, the polynomial –x³ + 4x² + 7x – 2 has four terms: –x³, 4x², 7x, and –2.
Each term has a coefficient:
- Coefficient of x³ is –1
- Coefficient of x² is 4
- Coefficient of x is 7
- Constant term –2 is the coefficient of x⁰, since x⁰ = 1.

Constant Polynomials

Examples include numbers like 2, –5, and 7.
The zero polynomial, represented as 0, is a constant polynomial with crucial importance in the study of polynomials.

Identifying Polynomials

Expressions such as x + 1/x and x + 3 are not polynomials due to negative exponents or fractions in the terms.
Only whole number exponents are valid in polynomial expressions.

Polynomial Notation

Polynomials can be denoted as p(x), q(x), r(x), etc.
Examples:
- p(x) = 2x² + 5x – 3
- q(x) = x³ – 1
- r(y) = y³ + y + 1
- s(u) = 2 – u – u² + 6u⁵

Finding Zeros of Polynomials

To find zeros of a polynomial, solve p(x) = 0.
Example: The linear polynomial p(x) = 2x + 1 has one zero at x = –1/2.

Key Observations about Zeros

A zero of a polynomial is not necessarily zero; for example, polynomials can have non-zero zeros.
Every linear polynomial has exactly one zero, while polynomials may have multiple zeros.

Exercises

Importance of evaluating polynomials at specific values (e.g., p(0), p(1), p(2)) to verify zeros.
Example evaluations:
- p(x) = x² – 2x, checking x = 2 and x = 0 both yield zeros.

Factorization of Polynomials

The Factor Theorem states that if p(a) = 0 for polynomial p(x), then (x – a) is a factor of p(x).
This relates to the Remainder Theorem, which indicates how polynomials can be factored based on their roots.

Applications

Recognizing polynomial structure is essential for solving equations, understanding graphs, and analyzing functions in advanced mathematics.

This quiz involves evaluating polynomial expressions at specific values and verifying whether given values are zeroes of the polynomials. Questions include finding p(0), p(1), p(2) for different polynomials and checking if certain values make the polynomial expression zero.