Alg 2 (12): Polynomial Remainder Theorem

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 ).