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Sagot :
To determine the end behavior of the polynomial function [tex]\( f(x) = -x^2 (1 - 2x) (x + 2) \)[/tex], follow these detailed steps:
1. Expand the polynomial:
Start by expanding the polynomial [tex]\( f(x) \)[/tex].
[tex]\[ f(x) = -x^2 (1 - 2x) (x + 2) \][/tex]
2. Simplify the inner terms:
Expand the product within the parentheses.
[tex]\[ (1 - 2x) (x + 2) \][/tex]
[tex]\[ = 1 \cdot x + 1 \cdot 2 + (-2x) \cdot x + (-2x) \cdot 2 \][/tex]
[tex]\[ = x + 2 - 2x^2 - 4x \][/tex]
Combine like terms:
[tex]\[ = -2x^2 - 3x + 2 \][/tex]
3. Combine with the outer term:
Now, multiply this result by [tex]\(-x^2\)[/tex]:
[tex]\[ f(x) = -x^2 \cdot (-2x^2 - 3x + 2) \][/tex]
[tex]\[ = (-x^2) \cdot (-2x^2) + (-x^2) \cdot (-3x) + (-x^2) \cdot 2 \][/tex]
[tex]\[ = 2x^4 + 3x^3 - 2x^2 \][/tex]
4. Identify the leading term:
The leading term (the term with the highest degree) of the polynomial is [tex]\( 2x^4 \)[/tex].
5. Determine the degree and the leading coefficient:
- The degree of the polynomial is 4 (since [tex]\( 2x^4 \)[/tex] is the term with the highest exponent).
- The leading coefficient is 2 (the coefficient of [tex]\( x^4 \)[/tex]).
6. Analyze the end behavior:
For a polynomial [tex]\( f(x) = ax^n \)[/tex]:
- If [tex]\( n \)[/tex] (the degree) is even and [tex]\( a \)[/tex] (the leading coefficient) is positive, as [tex]\( x \rightarrow \pm \infty \)[/tex], [tex]\( f(x) \rightarrow \infty \)[/tex].
- If [tex]\( n \)[/tex] is even and [tex]\( a \)[/tex] is negative, as [tex]\( x \rightarrow \pm \infty \)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex].
- If [tex]\( n \)[/tex] is odd and [tex]\( a \)[/tex] is positive, as [tex]\( x \rightarrow \infty\)[/tex], [tex]\( f(x) \rightarrow \infty \)[/tex] and as [tex]\( x \rightarrow -\infty\)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex].
- If [tex]\( n \)[/tex] is odd and [tex]\( a \)[/tex] is negative, as [tex]\( x \rightarrow \infty\)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex] and as [tex]\( x \rightarrow -\infty\)[/tex], [tex]\( f(x) \rightarrow \infty \)[/tex].
Given our polynomial [tex]\( f(x) = 2x^4 + 3x^3 - 2x^2 \)[/tex], we have:
- [tex]\( n = 4 \)[/tex] (even)
- [tex]\( a = 2 \)[/tex] (positive)
Therefore, the end behavior of the polynomial [tex]\( f(x) = -x^2 (1 - 2x) (x + 2) \)[/tex] is:
[tex]\[ f(x) \rightarrow -\infty \text{ as } x \rightarrow \pm\infty \][/tex]
Hence, the correct answer is:
- As [tex]\( x \rightarrow \infty, f(x) \rightarrow-\infty \)[/tex] and as [tex]\( x \rightarrow-\infty, f(x) \rightarrow -\infty \)[/tex].
1. Expand the polynomial:
Start by expanding the polynomial [tex]\( f(x) \)[/tex].
[tex]\[ f(x) = -x^2 (1 - 2x) (x + 2) \][/tex]
2. Simplify the inner terms:
Expand the product within the parentheses.
[tex]\[ (1 - 2x) (x + 2) \][/tex]
[tex]\[ = 1 \cdot x + 1 \cdot 2 + (-2x) \cdot x + (-2x) \cdot 2 \][/tex]
[tex]\[ = x + 2 - 2x^2 - 4x \][/tex]
Combine like terms:
[tex]\[ = -2x^2 - 3x + 2 \][/tex]
3. Combine with the outer term:
Now, multiply this result by [tex]\(-x^2\)[/tex]:
[tex]\[ f(x) = -x^2 \cdot (-2x^2 - 3x + 2) \][/tex]
[tex]\[ = (-x^2) \cdot (-2x^2) + (-x^2) \cdot (-3x) + (-x^2) \cdot 2 \][/tex]
[tex]\[ = 2x^4 + 3x^3 - 2x^2 \][/tex]
4. Identify the leading term:
The leading term (the term with the highest degree) of the polynomial is [tex]\( 2x^4 \)[/tex].
5. Determine the degree and the leading coefficient:
- The degree of the polynomial is 4 (since [tex]\( 2x^4 \)[/tex] is the term with the highest exponent).
- The leading coefficient is 2 (the coefficient of [tex]\( x^4 \)[/tex]).
6. Analyze the end behavior:
For a polynomial [tex]\( f(x) = ax^n \)[/tex]:
- If [tex]\( n \)[/tex] (the degree) is even and [tex]\( a \)[/tex] (the leading coefficient) is positive, as [tex]\( x \rightarrow \pm \infty \)[/tex], [tex]\( f(x) \rightarrow \infty \)[/tex].
- If [tex]\( n \)[/tex] is even and [tex]\( a \)[/tex] is negative, as [tex]\( x \rightarrow \pm \infty \)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex].
- If [tex]\( n \)[/tex] is odd and [tex]\( a \)[/tex] is positive, as [tex]\( x \rightarrow \infty\)[/tex], [tex]\( f(x) \rightarrow \infty \)[/tex] and as [tex]\( x \rightarrow -\infty\)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex].
- If [tex]\( n \)[/tex] is odd and [tex]\( a \)[/tex] is negative, as [tex]\( x \rightarrow \infty\)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex] and as [tex]\( x \rightarrow -\infty\)[/tex], [tex]\( f(x) \rightarrow \infty \)[/tex].
Given our polynomial [tex]\( f(x) = 2x^4 + 3x^3 - 2x^2 \)[/tex], we have:
- [tex]\( n = 4 \)[/tex] (even)
- [tex]\( a = 2 \)[/tex] (positive)
Therefore, the end behavior of the polynomial [tex]\( f(x) = -x^2 (1 - 2x) (x + 2) \)[/tex] is:
[tex]\[ f(x) \rightarrow -\infty \text{ as } x \rightarrow \pm\infty \][/tex]
Hence, the correct answer is:
- As [tex]\( x \rightarrow \infty, f(x) \rightarrow-\infty \)[/tex] and as [tex]\( x \rightarrow-\infty, f(x) \rightarrow -\infty \)[/tex].
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