Algebra II Section 3 5 Finding Real Roots
Algebra II Section 3. 5 Finding Real Roots of Polynomial Equations
Warm-up •
• When given roots, you can find factors and multiply them to get a polynomial function. • x=6 x=7 x= -1 • Factors (x-6) (x-7) (x+1) • f(x)= (x-6)(x-7)(x+1) Standard form?
Going backwards! roots→factors→function
Recall roots are values where the graph of a function crosses the x-axis.
• When the graph of a polynomial is tangent to the x-axis, the result is a double root • When the graph has a “valley” or a “hill” but doesn’t cross the x-axis, the result is imaginary roots.
• Sometimes you can find roots by factoring! • MAKE SURE ONE SIDE OF THE EQUATION EQUALS ZERO BEFORE YOU TRY TO FACTOR. • Try these • Pg. 186 #2 -7
Finding roots with the rational root theorem • A linear function has a maximum of one root • A quadratic function has a maximum of two roots • A cubic function has a maximum of three roots. • As the degree of the polynomial function increases by one degree, the graph has an additional turn resulting in one more possible root.
The food chain for numbers Complex Numbers Real Rational a+bi Imaginary Irrational Integers Whole Natural numbers that can be written as a fraction
q p • Rational root theorem (for polynomials with integer coefficients) every rational root can be written as p/q where p is a factor of the constant term and q is a factor of the leading coefficient
Who remembers how to do long division?
Dad, Mom, Brother, Sister As in Divide, Multiply, Bring Down, Subtract
There is a short cut method called synthetic division. • Write the possible root in a box followed by all the coefficients (include 0) where terms are missing in the polynomial • Bring down the leading coefficient • Multiply it by the number in the box and place it below the next coefficient • Add the numbers and repeat to the last coefficient • If the final result is a zero, you found a root!
• Before trying a different possible root, try the same one again with the quotient. This will check for a double root!
Try these! • Pg. 186 #11 -14
Classwork/homework Pg. 186 #15 -20, 27 a, find all the possible rational roots, 29, 32
- Slides: 27