512 64 18 16] 4 2 2 2 8 1 42 111 ( 8.1.2.1.1 4,2121 81111
1 answer:
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Answer:
Step-by-step explanation:
a) 1) Polynomial is closed under addition. In polynomial addition, only the coefficient change and variables and it powers does not change.
Example: Add 3x² - 2x + 7 and -2x² - 5x + 3
3x² - 2x + 7 - 2x² - 5x + 3 = 3x² -2x² - 2x - 5x + 7 + 3
= x² - 7x + 10
2)Polynomial is closed under addition. In polynomial subtraction, only the coefficient change and variables and it powers does not change.
Example: Subtract 3x - 2 from 4x + 2
4x + 2 - (3x - 2) = 4x + 2 - 3x + 2 = 7x + 4
3) Polynomial is closed under multiplication. In this, exponents of the variables are added according to the exponent rule. As the exponent of polynomials are whole numbers and whole numbers are closed under addition, the result will be a whole number.
-2x³ * 3x² =
4) Polynomial is not closed under multiplication. In division, we may get a negative exponent. So, it is not closed under division.
Addition subtraction multiplication Division
Natural numbers yes not closed Yes Not closed
Integers Yes Yes Yes not closed
Rational numbers Yes Yes Yes not closed
Irrational numbers not closed not closed not closed not closed
1) The function is 3(x + 2)³ - 3
2) The end behaviour is the limits when x approaches +/- infinity.
3) Since the polynomial is of odd degree you can predict that the ends head off in opposite direction. The limits confirm that.
4) The limit when x approaches negative infinity is negative infinity, then the left end of the function heads off downward (toward - ∞).
5) The limit when x approaches positive infinity is positivie infinity, then the right end of the function heads off upward (toward + ∞).
6) To graph the function it is important to determine:
- x-intercepts
- y-intercepts
- critical points: local maxima, local minima, and inflection points.
7) x-intercepts ⇒ y = 0
⇒ <span>3(x + 2)³ - 3 = 0 ⇒ (x + 2)³ - 1 = 0
</span>
<span>⇒ (x + 2)³ = -1 ⇒ x + 2 = 1 ⇒ x = - 1
</span>
8) y-intercepts ⇒ x = 0
y = <span>3(x + 2)³ - 3 = 3(0 + 2)³ - 3 = 0 - 3×8 - 3 = 24 - 3 = 21
</span><span>
</span><span>
</span><span>9) Critical points ⇒ first derivative = 0
</span><span>
</span><span>
</span><span>i) dy / dx = 9(x + 2)² = 0
</span><span>
</span><span>
</span><span>⇒ x + 2 = 0 ⇒ x = - 2
</span><span>
</span><span>
</span><span>ii) second derivative: to determine where x = - 2 is a local maximum, a local minimum, or an inflection point.
</span><span>
</span><span>
</span><span>y'' = 18 (x + 2); x = - 2 ⇒ y'' = 0 ⇒ inflection point.
</span><span>
</span><span>
</span><span>Then the function does not have local minimum nor maximum, but an inflection point at x = -2.
</span><span>
</span><span>
</span><span>Using all that information you can graph the function, and I attache the figure with the graph.
</span>
2) The end behaviour is the limits when x approaches +/- infinity.
3) Since the polynomial is of odd degree you can predict that the ends head off in opposite direction. The limits confirm that.
4) The limit when x approaches negative infinity is negative infinity, then the left end of the function heads off downward (toward - ∞).
5) The limit when x approaches positive infinity is positivie infinity, then the right end of the function heads off upward (toward + ∞).
6) To graph the function it is important to determine:
- x-intercepts
- y-intercepts
- critical points: local maxima, local minima, and inflection points.
7) x-intercepts ⇒ y = 0
⇒ <span>3(x + 2)³ - 3 = 0 ⇒ (x + 2)³ - 1 = 0
</span>
<span>⇒ (x + 2)³ = -1 ⇒ x + 2 = 1 ⇒ x = - 1
</span>
8) y-intercepts ⇒ x = 0
y = <span>3(x + 2)³ - 3 = 3(0 + 2)³ - 3 = 0 - 3×8 - 3 = 24 - 3 = 21
</span><span>
</span><span>
</span><span>9) Critical points ⇒ first derivative = 0
</span><span>
</span><span>
</span><span>i) dy / dx = 9(x + 2)² = 0
</span><span>
</span><span>
</span><span>⇒ x + 2 = 0 ⇒ x = - 2
</span><span>
</span><span>
</span><span>ii) second derivative: to determine where x = - 2 is a local maximum, a local minimum, or an inflection point.
</span><span>
</span><span>
</span><span>y'' = 18 (x + 2); x = - 2 ⇒ y'' = 0 ⇒ inflection point.
</span><span>
</span><span>
</span><span>Then the function does not have local minimum nor maximum, but an inflection point at x = -2.
</span><span>
</span><span>
</span><span>Using all that information you can graph the function, and I attache the figure with the graph.
</span>