MAC 1140 – Section 3.2
Division of Polynomials
NOTES:
A polynomial expression of degree n is written as: anxn + an-1xn-1 + …… + a2x2 + a1x + a0
where an is known as the leading coefficient and a0 is the constant term, for example:
7x5 – 3x3 + 5x – 9
is a fifth degree polynomial with leading coefficient 7 and constant of –9
A Linear Expression is a polynomial of degree 1 (mx + b can be written as a1x + a0)
A Quadratic expression is a polynomial of degree 2 (ax2 + bx + c can be rewritten as a2x2 + a1x + a0)
A polynomial expression of degree 3 is known as a cubic expression, and any degree greater than 3 is known as
a "higher order polynomial".
Dividing Polynomials Using Synthetic Division
1. Divide 3x3 – 5x2 – 3x + 4 by x – 2 using synthetic division.
dividing by a binomial. Write quotient and remainder.
You can only use synthetic division when
2. Practice: Use synthetic division to find the quotient and remainder when
x4 + 5x3 – 15x + 10 is divided by x + 3
Applications of Polynomial Division
1.
For 15  6, give the quotient and remainder. Quotient =
15 =
or 15 = 6  Q + R
Remainder =
In general, dividend = divisor  quotient + remainder
For problem 2 above, x4 + 5x3 – 15x + 10 = (x+3)  __________________ + ________________
If f(x) = x4 + 5x3 – 15x + 10 , then by substitution f(-3)= (-3)4+5(-3)3-15(-3)+10=________ .
2. Thus, the remainder theorem:
If R is the remainder when p(x) is divided by (x-c), then p(c) = R.
Given f(x) = 5x4 + 10x 3 – 12x2 – 25 use synthetic division and the remainder theorem to evaluate f(-3)
3. Factoring Polynomials using the Factor Theorem:
The number c is a zero of the polynomial function y =p(x) if and only if (x-c) is a factor of the
polynomial. For (x-c) to be a factor of the polynomial, the remainder must be zero when the
polynomial is divided synthetically by x -c.
Use synthetic division and the factor theorem to verify that the given number is a zero of the polynomial
function, and use your results to factor the polynomial. f(x) = x3 - 4x2 + 5x - 2
x=2
Zeros on the Graphing Calculator
4. Real zeros of a polynomial function are the x-intercepts.
Enter the function into Y1, 0 into Y2, and use the intersect option of 2nd Trace to determine the real zeros of
the function f(x) = 3x3 + 8x2 – 5x – 6
5. Determine EXACT VALUES for the real zeros
Use one of the integer roots from f(x) = 3x3 + 8x2 – 5x – 6 with synthetic division to reduce the x3 to an x2
expression. Then set the x2 polynomial = 0 and "solve" by factoring, square rooting, or using the quadratic
formula.
6. Determine EXACT VALUES for all real zeros of the polynomial: f(x) = x3 - 2x + 1
7. Determine EXACT VALUES for ALL ZEROS of the polynomial:
f(x) = x3 + 8x2 + 3x + 24
8. Determine EXACT VALUES for ALL ZEROS of the polynomial: f(x) = 4x4 -7x2 + 3