What is the number of distinct possible rational roots of p(x)=2x2+7x+6?
I need help on how to solve this
its meant to say p(x)=2x^2+7x+6
well a quadratic can have two roots but let's see
a = 2
b = 7
c = 6
x = [-b+/- sqrt(b^2-4ac)]/2a
if either is to be rational
sqrt(b^2-4ac) must be rational
sqrt (49 - 48) = sqrt 1 = 1
so ok, this will have 2 rational roots
they are
(-7 +1)/4 and(-7-1)/4
-3/2 and -2
===============
check
(x+ 3/2)(x+2) = x^2 + 7/2 x + 6/2 = 0
or 2 x^2 + 7 x + 6 = 0 ok
thank you
To determine the number of distinct possible rational roots for a polynomial equation, you can use the Rational Root Theorem. The theorem states that if a rational root exists for a polynomial equation with integer coefficients, it must have the form p/q, where p is a factor of the constant term (in this case, 6) and q is a factor of the leading coefficient (in this case, 2).
To solve this, follow these steps:
Step 1: Factor the constant term (6) and the leading coefficient (2). The factors of 6 are ±1, ±2, ±3, and ±6. The factors of 2 are ±1 and ±2.
Step 2: Write down all possible combinations of p/q using the factors you obtained in step 1. For example, if p is a factor of 6 and q is a factor of 2, combine them to get p/q.
Step 3: To find the possible rational roots, calculate p/q for each combination obtained in step 2. Simplify the fractions if possible.
Step 4: Substitute each possible rational root into the polynomial equation p(x) = 2x^2 + 7x + 6. If any of the substitutions result in p(x) equal to zero, then that value is a root of the equation.
Step 5: Count the number of distinct roots you found in step 4.
Applying these steps to the equation p(x) = 2x^2 + 7x + 6:
Step 1: The factors of 6 are ±1, ±2, ±3, and ±6. The factors of 2 are ±1 and ±2.
Step 2: Combine the factors obtained: ±1/1, ±2/1, ±3/1, and ±6/1
Step 3: Simplify the fractions: ±1, ±2, ±3, and ±6
Step 4: Substitute each possible rational root into the equation:
- p(±1) = 2(±1)^2 + 7(±1) + 6 = 2 - 7 + 6 = 1
- p(±2) = 2(±2)^2 + 7(±2) + 6 = 8 - 14 + 6 = 0 (This means ±2 is a root)
- p(±3) = 2(±3)^2 + 7(±3) + 6 = 18 - 21 + 6 = 3
- p(±6) = 2(±6)^2 + 7(±6) + 6 = 72 - 42 + 6 = 36 + 6 = 42
Step 5: As we found one distinct root (±2), the answer is 1.
Therefore, the number of distinct possible rational roots for the equation p(x) = 2x^2 + 7x + 6 is 1.