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  -3, 1, 4 are the x-intercepts

Step-by-step explanation:

The remainder theorem tells you that dividing a polynomial f(x) by (x-a) will result in a remainder that is the value of f(a). That remainder will be zero when (x-a) is a factor of f(x).

In terms of finding x-intercepts, this means we can reduce the degree of the polynomial by factoring out the factor (x-a) we found when we find a value of "a" that makes f(a) = 0.

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For the given polynomial, we notice that the sum of the coefficients is zero:

  1 -2 -11 +12 = 0

This means that x=1 is a zero of the polynomial, and we have found the first x-intercept point we can plot on the given number line.

Using synthetic division to find the quotient (and remainder) from division by (x-1), we see that ...

  f(x) = (x -1)(x² -x -12)

We know a couple of factors of 12 that differ by 1 are 3 and 4, so we suspect the quadratic factor above can be factored to give ...

  f(x) = (x -1)(x -4)(x +3)

Synthetic division confirms that the remainder from division by (x -4) is zero, so x=4 is another x-intercept. The result of the synthetic division confirms that x=-3 is the remaining x-intercept.

The x-intercepts of f(x) are -3, 1, 4. These are the points you want to plot on your number line.

step-by-step explanation: 4 cause 2+2=4