Transcript Solving Polynomials

Solving Polynomials
Solving Polynomials
• Factoring Options
1.
2.
3.
4.
5.
GCF Factoring (take-out a common term)
Sum or Difference of Cubes
Factor by Grouping
U Substitution
Polynomial Division (to factor out a binomial term)
Take-out common monomial
(GCF Factoring)
1)
3x  48 x
2)
6 x  8x  0
4
3
2
2
Sum or Difference of Cubes
1. Solve x 3  8  0
2. Solve x 3  27  0
Factor by Grouping
3)
x  3x  3x  9  0
4)
x 3  2 x 2  3x  6  0
3
2
Solving an Equation of Quadratic Type
(“U” Substitution)
5)
x  3x  2  0
4
2
6) x 4  5 x 2  36  0
Practice
• Polynomials WS
Synthetic Division
• Use synthetic division to find the quotient
and the remainder when 6 x3  19 x 2  16 x  4 is
divided by x – 2. If x – 2 is a factor, then
factor the polynomial completely.
• How can we determine whether x – 2 is a factor?
Factor Theorem
• A polynomial f(x) has a factor x – a iff the
remainder is 0.
Example 1
• Use synthetic division to determine
whether x – 1 is a factor of x³ - 1.
Example 2
• x = -4 is a solution of x³ - 28x – 48 = 0.
Use synthetic division to factor and find all
remaining solutions.
Example 3
• x + 3 is a factor of y = 3x³ + 2x² - 19x + 6.
Find all the zeros of this polynomial.
Rational Roots (Zeros) Test
• Every rational zero that is possible for a
given polynomial can be expressed as the
factors of the constant term divided by the
factors of the leading coefficient.
Example 4
• List all possible rational roots for the
polynomial y = 10x³ - 15x² - 16x + 12.
Then, divide out the factor and solve for all
remaining zeros.
Example 5
• List all possible rational roots for the
polynomial y = x³ - 7x – 6. Then, divide
out the factor and solve for all remaining
zeros.
Practice
• Pg. 213 (53 – 67 odd, 68)
• Pg. 278 (41, 43, 55, 57, 59)