5p + 3 = 15 please convert this into a WORD PROBLEM
2 answers:
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Given:
The leading coefficient of a polynomial is 8.
Polynomial roots are 1 and 2.
The graph passes through the point (4,5).
To find:
The 3rd root and the equation of the polynomial.
Solution:
The factor form of a polynomial is:
Where, a is a constant and are the roots of the polynomial.
Polynomial roots are 1 and 2. So, and
are the factors of the polynomial.
Let the third root of the polynomial by c, then is a factor of the polynomial.
The leading coefficient of a polynomial is 8. So, a=8 and the equation of the polynomial is:
The graph passes through the point (4,5). Putting , we get
Divide both sides by 48.
Therefore, the 3rd root on the polynomial is .
There is a multiple zero at 0 (which means that it touches there), and there are single zeros at -2 and 2 (which means that they cross). There is also 2 imaginary zeros at i and -i.
You can find this by factoring. Start by pulling out the greatest common factor, which in this case is -x^2.
-x^6 + 3x^4 + 4x^2
-x^2(x^4 - 3x^2 - 4)
Now we can factor the inside of the parenthesis. You do this by finding factors of the last number that add up to the middle number.
-x^2(x^4 - 3x^2 - 4)
-x^2(x^2 - 4)(x^2 + 1)
Now we can use the factors of two perfect squares rule to factor the middle parenthesis.
-x^2(x^2 - 4)(x^2 + 1)
-x^2(x - 2)(x + 2)(x^2 + 1)
We would also want to split the term in the front.
-x^2(x - 2)(x + 2)(x^2 + 1)
(x)(-x)(x - 2)(x + 2)(x^2 + 1)
Now we would set each portion equal to 0 and solve.
First root
x = 0 ---> no work needed
Second root
-x = 0 ---> divide by -1
x = 0
Third root
x - 2 = 0
x = 2
Forth root
x + 2 = 0
x = -2
Fifth and Sixth roots
x^2 + 1 = 0
x^2 = -1
x = +/-
x = +/- i