SOLUTION: a) Find all zeros of the polynomial algebraically {{{ P(x)=18x^4-21x^3-81x^2+84x+36 }}}. b) Then write the polynomial in factored form. c) Sketch the graph of P(x) showing all re

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: a) Find all zeros of the polynomial algebraically {{{ P(x)=18x^4-21x^3-81x^2+84x+36 }}}. b) Then write the polynomial in factored form. c) Sketch the graph of P(x) showing all re      Log On


   

Question 1155519: a) Find all zeros of the polynomial algebraically +P%28x%29=18x%5E4-21x%5E3-81x%5E2%2B84x%2B36+.
b) Then write the polynomial in factored form.
c) Sketch the graph of P(x) showing all real zeros, y-intercept, and end behavior.
Found 2 solutions by MathLover1, MathTherapy:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
a) Find all zeros of the polynomial algebraically
+P%28x%29=18x%5E4-21x%5E3-81x%5E2%2B84x%2B36+
b) Then write the polynomial in factored form.
P%28x%29=18x%5E4-21x%5E3-81x%5E2%2B84x%2B36
P%28x%29+=+3+%286+x%5E4+-+7+x%5E3+-+27+x%5E2+%2B+28+x+%2B+12%29
P%28x%29=3++%286+x%5E4+%2B+5+x%5E3+%2B17+x%5E2+-+6x-12x%5E3-10x%5E2-34x%2B12%29+

P%28x%29=3++%286x%5E3%28x+-2%29%2B+5x%5E2%28x-2%29+%2B+17x%28+x-2%29+-6+%28x-2%29%29+
P%28x%29=3+%28x+-+2%29%286+x%5E3+%2B+5+x%5E2+-+17+x+-+6%29+
P%28x%29=3+%28x+-+2%29+%286+x%5E3+%2B12+x%5E2-7x%5E2+-+14+x-3x+-+6%29+
P%28x%29=3+%28x+-+2%29+%28%286+x%5E3+%2B12+x%5E2%29-%287x%5E2+%2B+14+x%29-%283x+%2B+6%29%29
P%28x%29=3+%28x+-+2%29+%286x%5E2%28+x+%2B2+%29-7x%28x+%2B+2%29-3%28x+%2B+2%29%29
P%28x%29=3+%28x+-+2%29+%28+x+%2B2+%29%286x%5E2-7x-3%29
P%28x%29=3+%28x+-+2%29+%28+x+%2B2+%29%286x%5E2%2B2x-9x-3%29
P%28x%29=3+%28x+-+2%29+%28+x+%2B2+%29%28%286x%5E2%2B2x%29-%289x%2B3%29%29
P%28x%29=3+%28x+-+2%29+%28+x+%2B2+%29%282x%283x%2B1%29-3%283x%2B1%29%29
P%28x%29=3+%28x+-+2%29+%28x+%2B+2%29+%282x+-+3%29+%283x+%2B+1%29+
zeros:
if %28x+-+2%29=0=>x=2
if %28x+%2B+2%29=0=>x=-2
if %282x+-+3%29=0=>x=3%2F2
if %283x+%2B+1%29=0=>x=-1%2F3



c) Sketch the graph of P(x) showing all real zeros, y-intercept, and end behavior.



Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
a) Find all zeros of the polynomial algebraically +P%28x%29=18x%5E4-21x%5E3-81x%5E2%2B84x%2B36+.
b) Then write the polynomial in factored form.
c) Sketch the graph of P(x) showing all real zeros, y-intercept, and end behavior. 


Using the RATIONAL ROOT THEOREM, we find 2 of the zeroes of the function to be: - 2 and 2, thereby leading to factors: matrix%281%2C4%2C+%28x+%2B+2%29%28x+-+2%29%2C+and%2C+divisor%2C+x%5E2+-+4%29

Now, using the divisor: x%5E2+-+4 and LONG-DIVISION of POLYNOMIALS, we find the QUOTIENT of 6x%5E4+-+7x%5E3+-+27x%5E2+%2B+28x+%2B+12%29%2F%28x%5E2+-+4%29 to be: 6x%5E2+-+7x+-+3, which can be factored as: .

This gives us: (3x + 1)(2x - 3) = 0

3x + 1 = 0            or            2x - 3 = 0

3x = - 1              or            2x = 3



Therefore, zeroes of 

We ALSO see that the factors of: