Podcast
Questions and Answers
What does the Factor Theorem state about a polynomial and a factor of the form $(x - a)$?
What does the Factor Theorem state about a polynomial and a factor of the form $(x - a)$?
- $f(a) = a$ implies $(x - a)$ is a factor of $f(x)$
- $(x - a)$ is a factor if $f(a) \neq 0$
- $f(a) = 0$ implies $(x - a)$ is a factor of $f(x)$ (correct)
- $f(a) = 1$ implies $(x - a)$ is a factor of $f(x)$
In the polynomial $f(x) = x^2 - 5x + 6$, what will $f(2)$ evaluate to?
In the polynomial $f(x) = x^2 - 5x + 6$, what will $f(2)$ evaluate to?
- 2
- 4
- 0 (correct)
- 6
If $f(a) = 0$, which conclusion can be drawn regarding the polynomial and the factor?
If $f(a) = 0$, which conclusion can be drawn regarding the polynomial and the factor?
- $f(x)$ has no real roots
- $(x - a)$ is not a factor of $f(x)$
- $(x - a)$ is surely a factor of $f(x)$ (correct)
- $f(x)$ has multiple factors
If $(x - 2)$ is a factor of a polynomial, what can be inferred about the polynomial when $x = 2$?
If $(x - 2)$ is a factor of a polynomial, what can be inferred about the polynomial when $x = 2$?
What is the result of factoring the polynomial $f(x) = x^2 - 5x + 6$ given that $(x - 2)$ is a factor?
What is the result of factoring the polynomial $f(x) = x^2 - 5x + 6$ given that $(x - 2)$ is a factor?
How does the Factor Theorem differ from the Remainder Theorem?
How does the Factor Theorem differ from the Remainder Theorem?
What is the implication when $f(a) \neq 0$ for a polynomial $f(x)$?
What is the implication when $f(a) \neq 0$ for a polynomial $f(x)$?
What is the primary use of the Factor Theorem in algebra?
What is the primary use of the Factor Theorem in algebra?
Flashcards
Factor Theorem
Factor Theorem
A theorem stating a polynomial f(x) has (x-a) as a factor if f(a) = 0.
Remainder Theorem
Remainder Theorem
If a polynomial f(x) is divided by (x-a), the remainder is f(a).
Factor
Factor
A polynomial expression that can multiply with another to form a polynomial.
Polynomial
Polynomial
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Substitution in polynomials
Substitution in polynomials
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Example polynomial
Example polynomial
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Zero of a polynomial
Zero of a polynomial
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Factoring a polynomial
Factoring a polynomial
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Study Notes
Factor Theorem
- The Factor Theorem is a special case of the Remainder Theorem.
- It helps determine if a polynomial has a specific factor.
- If a polynomial 𝑓(𝑥) is divided by (𝑥 − 𝑎), then (𝑥 − 𝑎) is a factor of 𝑓(𝑥) if and only if 𝑓(𝑎) = 0.
- This means if substituting 'a' into the polynomial gives a result of zero (𝑓(𝑎) = 0), then (𝑥 − 𝑎) is a factor.
- Conversely, if (𝑥 − 𝑎) is a factor, substituting 'a' into the polynomial will result in zero.
Example
- Consider the polynomial 𝑓(𝑥) = 𝑥² − 5𝑥 + 6.
- To determine if (𝑥 − 2) is a factor, substitute 𝑥 = 2 into the polynomial:
- 𝑓(2) = (2)² − 5(2) + 6 = 4 − 10 + 6 = 0
- Since 𝑓(2) = 0, (𝑥 − 2) is a factor of 𝑓(𝑥).
- Factoring the polynomial:
- 𝑓(𝑥) = (𝑥 − 2)(𝑥 − 3)
Application
- The Factor Theorem is helpful in finding factors of polynomials and solving polynomial equations.
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