Podcast
Questions and Answers
If a polynomial is divisible by (x - a), what can you conclude?
If a polynomial is divisible by (x - a), what can you conclude?
P(a) = 0
What can you conclude if P(a) = 0 for polynomial P(x)?
What can you conclude if P(a) = 0 for polynomial P(x)?
a is a zero/root/solution of P(x), (x - a) is a factor, P(x) is divisible by (x - a)
If a polynomial divided by (x - a) has remainder R, what can you conclude?
If a polynomial divided by (x - a) has remainder R, what can you conclude?
P(a) = R
What can you conclude if P(a) = b where b ≠ 0 for polynomial P(x)?
What can you conclude if P(a) = b where b ≠ 0 for polynomial P(x)?
Signup and view all the answers
Study Notes
Polynomial Remainder Theorem
- A polynomial ( P(x) ) is divisible by ( (x - a) ) if and only if ( P(a) = 0 ). This indicates that ( a ) is a zero, root, or solution of the polynomial.
- When ( (x - a) ) is a factor of ( P(x) ), it follows that ( P(x) ) will produce no remainder when divided by ( (x - a) ).
- If ( a ) is substituted into ( P(x) ) and results in zero, ( (x - a) ) can be considered a factor of ( P(x) ).
Implications of ( P(a) = 0 )
- If ( P(a) = 0 ), then ( a ) serves as a zero/root/solution of the polynomial ( P(x) ).
- The factor ( (x - a) ) confirms that ( P(x) ) is divisible by this expression.
- There will be no remainder when performing polynomial long division of ( P(x) ) by ( (x - a) ).
Remainder When Dividing Polynomials
- When dividing a polynomial ( P(x) ) by ( (x - a) ) yields a remainder ( R ), we find ( P(a) = R ).
- For instance, dividing ( 5x^2 - 4x + 1 ) by ( (x - 2) ) results in a remainder of 13, indicating that ( (x - 2) ) is not a factor of ( P(x) ).
- If substituting ( a ) into ( P(x) ) provides a value other than zero, ( (x - a) ) cannot be a factor of ( P(x) ).
Implications of Non-Zero Remainders
- If ( P(a) = b ) where ( b \neq 0 ), then ( a ) is not a zero/root/solution of the polynomial ( P(x) ).
- Consequently, ( (x - a) ) cannot be considered a factor of ( P(x) ), confirming that ( P(x) ) is not divisible by ( (x - a) ).
- When dividing ( P(x) ) by ( (x - a) ), there will be a non-zero remainder ( b ).
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of the Polynomial Remainder Theorem with interactive flashcards. This quiz explores key concepts including divisibility and evaluating polynomials at specific points. Perfect for Algebra 2 students looking to reinforce their learning.
