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Answer :
Sure! Let's work through this polynomial step-by-step to understand it better:
The given polynomial is:
[tex]\[ P(x) = -3x^3 - 19x^2 + 27x - 7 \][/tex]
Here are the steps to explore this polynomial:
1. Finding the Roots:
- Roots of a polynomial are the values of [tex]\( x \)[/tex] for which the polynomial equals zero.
- Solving the equation [tex]\( -3x^3 - 19x^2 + 27x - 7 = 0 \)[/tex] will give us the roots.
2. Differentiation:
- Finding the derivative of the polynomial helps us understand its behavior. The derivative of [tex]\( P(x) \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[ P'(x) = -9x^2 - 38x + 27 \][/tex]
- This derivative can indicate the slope of the tangent to the curve at any point [tex]\( x \)[/tex].
3. Evaluating at a Specific Point:
- To evaluate the polynomial at [tex]\( x = 1 \)[/tex], we substitute 1 for [tex]\( x \)[/tex] in the polynomial:
[tex]\[ P(1) = -3(1)^3 - 19(1)^2 + 27(1) - 7 = -3 - 19 + 27 - 7 = -2 \][/tex]
- Therefore, the value of the polynomial at [tex]\( x = 1 \)[/tex] is -2.
By performing the root calculation, differentiation, and evaluation, we gain valuable insights into the behavior and characteristics of the polynomial. The roots and derivative are essential in analyzing the turning points and the shape of the graph.
The given polynomial is:
[tex]\[ P(x) = -3x^3 - 19x^2 + 27x - 7 \][/tex]
Here are the steps to explore this polynomial:
1. Finding the Roots:
- Roots of a polynomial are the values of [tex]\( x \)[/tex] for which the polynomial equals zero.
- Solving the equation [tex]\( -3x^3 - 19x^2 + 27x - 7 = 0 \)[/tex] will give us the roots.
2. Differentiation:
- Finding the derivative of the polynomial helps us understand its behavior. The derivative of [tex]\( P(x) \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[ P'(x) = -9x^2 - 38x + 27 \][/tex]
- This derivative can indicate the slope of the tangent to the curve at any point [tex]\( x \)[/tex].
3. Evaluating at a Specific Point:
- To evaluate the polynomial at [tex]\( x = 1 \)[/tex], we substitute 1 for [tex]\( x \)[/tex] in the polynomial:
[tex]\[ P(1) = -3(1)^3 - 19(1)^2 + 27(1) - 7 = -3 - 19 + 27 - 7 = -2 \][/tex]
- Therefore, the value of the polynomial at [tex]\( x = 1 \)[/tex] is -2.
By performing the root calculation, differentiation, and evaluation, we gain valuable insights into the behavior and characteristics of the polynomial. The roots and derivative are essential in analyzing the turning points and the shape of the graph.
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