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Chapter 2: Problem 81

Determine whether the statement is true or false. Justify your answer. The zeros of the polynomial \(x^{3}-2 x^{2}-11 x+12 \geq 0\) divide the real number line into four test intervals.

Short Answer

Expert verified
The statement is true. The zeros of the polynomial \(x^{3}-2 x^{2}-11 x+12\) do divide the real number line into four test intervals.

Step by step solution

01

Finding the zeros

Zeros of the polynomial \(x^{3}-2x^{2}-11x+12\) are found by setting this polynomial equal to zero and solving for \(x\). \nSo, \(x^{3}-2x^{2}-11x+12=0\). Factorizing the polynomial, we get: \((x-1)(x-3)(x+4)=0\). So the zeroes are \(x=1\), \(x=3\) and \(x=-4\)
02

Check the number of intervals

Observe the real number line and mark points -4, 1 and 3. This divides the real number line into 4 intervals: \(-\infty\), -4, -4 to 1, 1 to 3 and 3 to \(\infty\).
03

Conclusion

So, the statement 'The zeros of the polynomial \(x^{3}-2 x^{2}-11 x+12 \geq 0\) divide the real number line into four test intervals.' is true.

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