Use factor theorem to determine the value of k for which x+2 is a factor of (x+1)^7+(2 x+k)^3
Answers
Factor theorem is a theorem linking factors and zeros of a polynomial. It is a special case of the polynomial remainder theorem.
The factor theorem states that a polynomial f(x) has a factor (x − k) if and only if f(k
For any given expression,find its factors manually and substitute those values in the variables.
For example,,
You wish to find the factors of
x^3 + 7x^2 + 8x + 2
Find the factors of 2 ie. 1,-1,2,-2.
Substitute those values in the variable and it should equal zero.
To do this you would use trial and error to find the first x value that causes the expression to equal zero. To find out if (x − 1) is a factor, substitute x = 1 into the polynomial above:
x^3 + 7x^2 + 8x + 2 = (1)3 + 7(1)2 + 8(1) + 2
= 1 + 7 + 8 + 2
= 18.
As this is equal to 18 and not 0 this means (x − 1) is not a factor of x^3 + 7x^2 + 8x + 2. So, we next try (x + 1) (substituting x = − 1 into the polynomial):
( − 1)3 + 7( − 1)2 + 8( − 1) + 2.
This is equal to 0. Therefore x = ( − 1), which is to say x + 1, is a factor, and − 1 is a root of x^3 + 7x^2 + 8x + 2.
The next two roots can be found by algebraically dividing x^3 + 7x^2 + 8x + 2 by (x + 1) to get a quadratic, which can be solved directly, by the factor theorem or by the quadratic equation.
and therefore (x + 1) and x^2 + 6x + 2 are the factors of x^3 + 7x^2 + 8x + 2.
Hope this helped.
Answer:
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