Section 2-4 Real Zeros of Polynomial Functions

Section 2-4 long division and the division algorithm the remainder and factor theorems reviewing the fundamental connection for polynomial functions synthetic division rational zeros theorem upper and lower bounds

Long Division long division for polynomials is just like long division for numbers it involves a dividend divided by a divisor to obtain a quotient and a remainder the dividend is the numerator of a fraction and the divisor is the denominator

Division Algorithm if f(x) is the dividend, d(x) is the divisor, q(x) the quotient, and r(x) the remainder, then the division algorithm can be stated two ways

Remainder Theorem if a polynomial f (x) is divided by x – k, then the remainder is f (k) in other words, the remainder of the division problem would be the same value as plugging in k into the f (x) we can find the remainders without having to do long division later, we will find f (k) values without having to plug k into the function using a shortcut for long division

Factor Theorem the useful aspect of the remainder theorem is what happens when the remainder is 0 since the remainder is 0, f (k) = 0 which means that k is a zero of the polynomial it also means that x – k is a factor of the polynomial if we could find out what values yield remainders of 0 then we can find factors of polynomials of higher degree

Fundamental Connection k is a solution (or root) of the equation f (x) = 0 k is a zero of the function f (x) k is an x-intercept of the graph of f (x) x – k is a factor of f (x) For a real number k and a polynomial function f (x), the following statements are equivalent

Synthetic Division finding zeros and factors of polynomials would be simple if we had some easy way to find out which values would produce a remainder of 0 (long division takes too long) synthetic division is just that shortcut it allows us to quickly divide a function f (x) by a divisor x – k to see if it yields a remainder of 0

Synthetic Division it follows the same steps as long division without having to write out the variables and other notation it is really fast and easy if a zero is found, the resulting quotient is also a factor, and it is called the depressed equation because it will be one degree less than the original function

Synthetic Division

Synthetic Division

Synthetic Division The remainder is 0 so x – 3 is a factor and the quotient, 2x 2 + 3x + 4, is also a factor

Rational Zeros Theorems if you want to find zeros, you need to have an idea about which values to test in S.D. (synthetic division) the rational zeros theorem provides a list of possible rational zeros to test in S.D. they will be a value where, p must be a factor of the constant q must be a factor of the leading term

Finding Possible Rational Zeros

Upper and Lower Bounds a number k is an upper bound if there are no zeros greater than k; if k is plugged into S.D., the bottom line will have no sign changes a number k is a lower bound if there are no zeros less than k; if k is plugged into S.D., the bottom line will have alternating signs (0 can be considered + or -)

Upper and Lower Bounds if you are looking for zeros and you come across a lower bound, do not try any numbers less than that number if you are trying to find zeros and you come across an upper bound, do not try any numbers greater than that number