Question
True or False: -4 is a root \( \left(x\right. \)-int) of \( x^{3}-2 x^{2}+5 x+8 \)
Ask by Matthews Gross. in the United States
Feb 02,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
-4 is not a root of the polynomial \( x^{3} - 2x^{2} + 5x + 8 \).
Solution
**False.**
To determine whether \(-4\) is a root of the polynomial \( f(x) = x^{3} - 2x^{2} + 5x + 8 \), substitute \(-4\) into the polynomial and evaluate:
\[
f(-4) = (-4)^3 - 2(-4)^2 + 5(-4) + 8
\]
Calculate each term:
1. \((-4)^3 = -64\)
2. \(-2(-4)^2 = -2 \times 16 = -32\)
3. \(5(-4) = -20\)
4. \(+8\)
Now, add them together:
\[
-64 - 32 - 20 + 8 = -108
\]
Since \( f(-4) = -108 \neq 0 \), \(-4\) is **not** a root of the polynomial.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To find out if -4 is a root of the polynomial \( x^{3}-2 x^{2}+5 x+8 \), we substitute \( x = -4 \) into the equation: \[ (-4)^3 - 2(-4)^2 + 5(-4) + 8 = -64 - 32 - 20 + 8 = -108 \] Since the result is not equal to zero, -4 is not a root of the polynomial. So the statement is false. As you can see, polynomial equations can sometimes lead to surprising results! When checking roots, plugging in values is the quickest way to confirm whether a candidate is actually a solution. The lesson here? Always double-check your calculations to avoid misjudgments, especially when playing with negative numbers, because they have a sneaky habit of throwing you off!
