cubic polynomial p(x)=x^3-7x^2+14x-8 has ZEROS 1 and 2. whats the third zero?

Question

i do not know the meaning of ZEROS 1 and 2? care to explain ?

Answer 1

That means that p(1) = 0 and p(2) = 0. ("Zeroes" is another name for "roots".) It also means that (x - 1) and (x - 2) are factors of p(x). Hopefully, you know how to do synthetic division. You can divide (x - 1) into (x^3 - 7x^2 + 14x - 8), and then divide (x - 2) into the quotient (synthetically or by inspection) to get a binomial of the form (x - n), where n is the third zero.

You can also note that (x - 1)(x - 2) = x^2 - 3x + 2, which has a constant term of 2, and that the last term of p(x) is -8. You must multiply 2 by -4 to get -8, so the third factor should be (x - 4), giving you a zero of 4. And, indeed, a quick calculation will show that p(4) = 0.

Answer 2

Imagine that you plot a graph of the polynomial in the x and y axes on a graph paper. The points on the x-axis where the values of y are zero are then the zeros of the polynomial.
We first try to factorize the polynomial, using the two zero points given: x^3 - 7 x^2 + 14 x - 8 = 0 ...........1)
(x - 1 )( x - 2) = x^2 - 3x + 2. .......................2)
We look for another factor in the form (x - m) which, when multiplied by equation 2 will give equation 1. From visual inspection, the value of m must be 4 to satisfy this relation:
(x^2 - 3x + 2)(x - 4) = x^3 -3x^2 + 2x - 4x^2 + 12x -8
= x^3 - 7x^2 + 14x - 8
The 3rd zero of the polynomial, therefore, is x = 4

Answer 3

So the question is asking what is the third value for x that will make p(x) = 0. Given that you know 1 and 2 work you know that (x-1)(x-2)(x - P) = 0
so
x^2 - 3x +2(x-P) = 0
You can then see to get -8 in the final equation P must 4 as your third zero. Check
Does 4^3 - 7(4^2) + 14(4) - 8 = 0 ??
64 - 7(16) + 56 - 8 = 0 ??
64 - 112 + 56 - 8 = 0 YES!!

Answer 4

The "zeros 1 and 2" means that when x=0 or x=1, y=0.

Given that 1 and 2 are zeros, it follows that p(x) can be factored into the form (x-1)(x-2)(q(x)) = 0.
Then we just have to figure out what q(x) is.

Since the first term of p(x) is x^3, it follows that q(x) has an x in it.
Since the last term of p(x) is -8, it follows that q(x) has a -4 in it. This must be the case, since you have to get a -8 in the units term when you multiply out the parentheses, and (-1)*(-2)*(-4) = - 8.

Hence, p(x) = (x-1)(x-2)(x-4), and the third zero is 4.

Answer 5

If 1 is a zero then (x-1) is a factor of the equation

so you have (x-1)(x-2)(ax+b)

The last number has to be made up of -1*-2*b so b=-4
Since the coefficient of x^3 is 1 then a=1

factorised you have (x-1)(x-2)(x-4)

So the remaining root to be found is when x = 4.

Answer 6

zeros mean the root , or that value of x for which p(x) = 0

p(1) = 0
p(2) = 0

factorize p(x) as p(x) = (x-1)(x-2)(x-4)

So the third zero is 4 Answer is 4