x = 2/7 is a root of 49x^3 - 126x^2 + 60x - 8 = 0. Factor the polynomial completely and list all real solutions.
My answer: (x - 2)(7x - 2)^2 = 0 factored, and the solutions are x = 2 and x = 2/7.
looks good to me.
You can always check your answer by typing in the polynomial at wolframalpha.com
To factor the polynomial completely, we start with the given factor x = 2/7. We know that if x = 2/7 is a root of the polynomial, then x - 2/7 must be a factor of the polynomial.
First, let's divide the polynomial by x - 2/7 using polynomial long division:
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x - 2/7 | 49x^3 - 126x^2 + 60x - 8
- (49x^2 - 2x)
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- 128x^2 + 62x - 8
- (- 128x^2 + 4/7x)
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(61 94/7) x - 8
Now, we have the quotient as (61x - 94)/7 and the remainder as -8.
Therefore, we can rewrite the original polynomial as:
( x - 2/7 ) * ( 49x^2 - 2x - (61x - 94)/7)
Now, let's factor the quadratic polynomial 49x^2 - 2x - (61x - 94)/7. We can do this by factoring out the greatest common factor (in this case, 1/7) and then factoring the resulting quadratic expression:
( 49x^2 - 2x - (61x - 94)/7 )
= (1/7) * ( 343x^2 - 14x - 61x + 94 )
= (1/7) * ( 343x^2 - 75x + 94 )
Now, we need to factor the quadratic expression 343x^2 - 75x + 94. To do this, we can use the quadratic formula or factor it by grouping. However, upon inspection, we can see that this quadratic doesn't factor nicely, which means it doesn't have rational roots.
Therefore, the factored form of the polynomial is:
( x - 2/7 ) * ( 49x^2 - 2x - (61x - 94)/7 )
But since the quadratic factor doesn't factor further, the polynomial is already fully factored.
The solutions to the equation are x = 2/7 and the solutions to the quadratic 49x^2 - 2x - (61x - 94)/7 can be found by using the quadratic formula or by approximation methods.