Explanation Procedure used: The graph of a polynomial can sketch as follows. 1. Use the leading term test to determine the end behavior. 2. Find the zeros of the function by solving f ( x ) = 0 . Any real zeros are the first coordinates of the x -intercepts. 3. Use the x -intercepts (zeros) to divide the x -axis into the intervals, and choose a test point in each interval to determine the sign of all function values in that interval. 4. Find f ( 0 ) . This gives the y -intercept of the function. 5. If necessary, find additional function values to determine the general shape of the graph and then draw the graph. 6. As a partial check, use the facts that the graph has at most n − x intercepts and at most n − 1 turning points. Multiplicity of zeros can also be considered in order to check where the graph crosses or is tangent to the x axis. We can also check with a graphing calculator. Result used: The Factor Theorem: Let f ( x ) be a polynomial. a. If f ( c ) = 0 , then x − c is a factor of f ( x ) . b. If x − c is a factor of f ( x ) , then f ( c ) = 0 . Calculation: The given polynomial function is f ( x ) = x 4 + x 3 − 3 x 2 − 5 x − 2 . Use the leading term test to determine the end behavior. The leading term in the given polynomial is x 4 . So, the degree of the polynomial is 4, which is an even number. The coefficient of the leading term is 1, which is a positive number. So, it is clear that the value of f ( x ) → ∞ as x → ∞ and f ( x ) → − ∞ as x → − ∞ . Obtain the zeros of the function by solving f ( x ) = 0 . Any real zeros are the first coordinates of the x -intercepts. f ( x ) = 0 x 4 + x 3 − 3 x 2 − 5 x − 2 = 0 Find the zeros of the above polynomial by using synthetic division. Check x = − 1 is the zero of the polynomial. − 1 1 1 − 3 − 5 − 2 − 1 0 3 2 _ 1 0 − 3 − 2 0 From the above synthetic division, f ( − 1 ) = 0 . By the factor theorem, x + 1 is a factor of the polynomial f ( x ) . The function f ( x ) becomes f ( x ) = ( x + 1 ) ( x 3 − 3 x − 2 ) . To find the zero, solve f ( x ) = 0 . ( x + 1 ) ( x 3 − 3 x − 2 ) = 0 x = − 1 or x 3 − 3 x − 2 = 0 Check whether x = − 1 is the zero of the polynomial x 3 − 3 x − 2 = 0 . − 1 1 0 − 3 − 2 − 1 1 2 _ 1 − 1 − 2 0 From the above synthetic division, f ( − 1 ) = 0

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Chapter 4.3, Problem 50E

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To graph: The polynomial function f(x)=x4+x3−3x2−5x−2.

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Chapter 4.3, Problem 49E

Chapter 4.3, Problem 51E

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MODELING REAL LIFE Your checking account has a constant balance of $500. Let the function $m$ represent the balance of your savings account after $t$ years. The table shows the total balance of the accounts over time. Year, $t$ Total balance 0 1 2 3 4 5 $2500 $2540 $2580.80 $2622.42 $2664.86 $2708.16 a. Write a function $B$ that represents the total balance after $t$ years. Round values to the nearest hundredth, if necessary. $B\left(t\right)=$ Question 2 b. Find $B\left(8\right)$ . About $ a Question 3 Interpret $B\left(8\right)$ . b represents the total balance checking and saving accounts after 8 years the balance would be 16 / 10000 Word Limit16 words written of 10000 allowed Question 4 c. Compare the savings account to the account, You deposit $9000 in a savings account that earns 3.6% annual interest compounded monthly. A = 11998.70 SINCE 9000 is the principal ( 1+0.036/12)12 times 8 gives me aproxtimately 1997 14 / 10000 Word Limit14 words written of 10000 allowed Skip to…

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Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 2E Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - Determine the leading term, the leading...Ch. 4.1 - Prob. 10E

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To graph: The polynomial function f ( x ) = x 4 + x 3 − 3 x 2 − 5 x − 2 .

Solution Summary: The author explains how the graph of a polynomial can sketch as follows: Use the leading term test to determine the end behavior and find the zeros of the function.

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To graph: The polynomial function f(x)=x4+x3−3x2−5x−2.

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