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Brandy is asked to sketch a graph of m(x)=(2x+5)(x−3)3(8x+1)5

. How many turning points should her sketch include?(1 point)
Responses

10 at most
10 at most

two at most
two at most

eight at most
eight at most

seven at most

11 answers

To determine the number of turning points of the function \( m(x) = (2x+5)(x-3)^3(8x+1)^5 \), we first need to analyze the degree of the polynomial.

  1. Degree of each factor:

    • \( (2x+5) \) has degree 1.
    • \( (x-3)^3 \) has degree 3.
    • \( (8x+1)^5 \) has degree 5.

  2. Total degree of the polynomial: \[ \text{Total degree} = 1 + 3 + 5 = 9. \]

The maximum number of turning points for a polynomial is given by the degree of the polynomial minus 1. Therefore, the maximum number of turning points for the polynomial \( m(x) \) is:

\[ 9 - 1 = 8. \]

Thus, Brandy's sketch should include at most 8 turning points.

Final Answer: eight at most

Divide the polynomials 4x4+4x−10
by 2x2−3
to determine the quotient and remainder.(1 point)
Responses

The quotient is 2x2−3
. The remainder is 4x+1
.
The quotient is 2 x squared minus 3. The remainder is 4 x plus 1.

The quotient is 2x2−3
. The remainder is 4x−19
.
The quotient is 2 x squared minus 3. The remainder is 4 x minus 19.

The quotient is 2x2+3
. The remainder is 4x−1
.
The quotient is 2 x squared plus 3. The remainder is 4 x minus 1.

The quotient is 2x2+3
. The remainder is 4x−19
.

To perform the polynomial long division of \( 4x^4 + 4x - 10 \) by \( 2x^2 - 3 \), we follow these steps:

  1. Set Up the Division: We want to divide \( 4x^4 + 0x^3 + 4x - 10 \) by \( 2x^2 - 3 \).

  2. Divide the leading terms: \[ \frac{4x^4}{2x^2} = 2x^2 \] This will be the first term of the quotient.

  3. Multiply: Multiply \( 2x^2 \) by \( 2x^2 - 3 \): \[ 2x^2(2x^2 - 3) = 4x^4 - 6x^2 \]

  4. Subtract: Subtract \( (4x^4 - 6x^2) \) from the original polynomial: \[ (4x^4 + 0x^3 + 4x - 10) - (4x^4 - 6x^2) = 0x^4 + 0x^3 + 6x^2 + 4x - 10 \] This simplifies to: \[ 6x^2 + 4x - 10 \]

  5. Repeat: Now, divide the new leading term \( 6x^2 \) by \( 2x^2 \): \[ \frac{6x^2}{2x^2} = 3 \]

  6. Multiply: Multiply \( 3 \) by \( 2x^2 - 3 \): \[ 3(2x^2 - 3) = 6x^2 - 9 \]

  7. Subtract: Now subtract \( (6x^2 - 9) \) from \( 6x^2 + 4x - 10 \): \[ (6x^2 + 4x - 10) - (6x^2 - 9) = 0x^2 + 4x - 1 \] This simplifies to: \[ 4x - 1 \]

  8. Conclusion: Since the degree of the remainder \( 4x - 1 \) (which is 1) is less than the degree of the divisor \( 2x^2 - 3 \) (which is 2), we stop here.

So, the results are:

  • Quotient: \( 2x^2 + 3 \)
  • Remainder: \( 4x - 1 \)

The correct response is: The quotient is \( 2x^2 + 3 \). The remainder is \( 4x - 1 \).

Given that f(x)=9x2−50
, apply the Remainder Theorem to find the remainder of (9x2−50)÷(x−73)
.(1 point)
Responses

−29
negative 29

−1
negative 1

−50
negative 50

−2033

To find the remainder of the polynomial \( f(x) = 9x^2 - 50 \) when divided by \( x - 73 \) using the Remainder Theorem, we need to evaluate \( f(73) \).

The Remainder Theorem states that the remainder of a polynomial \( f(x) \) divided by \( x - c \) is \( f(c) \).

  1. Substitute \( x = 73 \) into \( f(x) \): \[ f(73) = 9(73)^2 - 50 \]

  2. Calculate \( 73^2 \): \[ 73^2 = 5329 \]

  3. Multiply by 9: \[ 9 \times 5329 = 47961 \]

  4. Subtract 50: \[ f(73) = 47961 - 50 = 47911 \]

Therefore, the remainder when \( 9x^2 - 50 \) is divided by \( x - 73 \) is not one of the options given.

However, to find the result consistent with the available responses, let's double-check:

  1. Calculate directly: \[ f(73) = 9(5329) - 50 = 47961 - 50 = 47911 \]

Since none of the listed options (-29, -1, -50, and -2033) corresponds to this value, please clarify if you intended a different divisor or polynomial. The correct remainder for the given calculation is \( 47911 \).

Given f(x)=2x3−5x2−68x+35
, apply the Factor Theorem and determine which of the following is a factor of f(x)
.(1 point)
Responses

(x+7)
left parenthesis x plus 7 right parenthesis

(2x+1)
left parenthesis 2 x plus 1 right parenthesis

(x+5)
left parenthesis x plus 5 right parenthesis

(x−3)

To determine which of the given expressions is a factor of the polynomial \( f(x) = 2x^3 - 5x^2 - 68x + 35 \), we can use the Factor Theorem. According to the Factor Theorem, \( (x - c) \) is a factor of \( f(x) \) if and only if \( f(c) = 0 \).

We'll evaluate \( f(x) \) for the roots indicated by the factors:

  1. Evaluate \( f(-7) \) for \( (x + 7) \): \[ f(-7) = 2(-7)^3 - 5(-7)^2 - 68(-7) + 35 \] \[ = 2(-343) - 5(49) + 476 + 35 \] \[ = -686 - 245 + 476 + 35 \] \[ = -686 - 245 + 511 \] \[ = -420 \quad (\text{Not a factor}) \]

  2. Evaluate \( f\left(-\frac{1}{2}\right) \) for \( (2x + 1) \) (which corresponds to \( x = -\frac{1}{2} \)): \[ f\left(-\frac{1}{2}\right) = 2\left(-\frac{1}{2}\right)^3 - 5\left(-\frac{1}{2}\right)^2 - 68\left(-\frac{1}{2}\right) + 35 \] \[ = 2\left(-\frac{1}{8}\right) - 5\left(\frac{1}{4}\right) + 34 + 35 \] \[ = -\frac{1}{4} - \frac{5}{4} + 34 + 35 \] \[ = -\frac{6}{4} + 69 \] \[ = -\frac{3}{2} + 69 \] \[ = 66.5 \quad (\text{Not a factor}) \]

  3. Evaluate \( f(-5) \) for \( (x + 5) \): \[ f(-5) = 2(-5)^3 - 5(-5)^2 - 68(-5) + 35 \] \[ = 2(-125) - 5(25) + 340 + 35 \] \[ = -250 - 125 + 340 + 35 \] \[ = -375 + 375 \] \[ = 0 \quad (\text{This is a factor}) \]

  4. Evaluate \( f(3) \) for \( (x - 3) \): \[ f(3) = 2(3)^3 - 5(3)^2 - 68(3) + 35 \] \[ = 2(27) - 5(9) - 204 + 35 \] \[ = 54 - 45 - 204 + 35 \] \[ = 9 - 204 + 35 \] \[ = 9 - 169 \] \[ = -160 \quad (\text{Not a factor}) \]

After evaluating \( f(x) \) for all the given options, we find that:

The factor of \( f(x) \) is \( (x + 5) \).

Given that x=2
is a zero of f(x)=x3−3x2−10x+24
, apply the Factor Theorem and factor f(x)
into linear factors.(1 point)
Responses

f(x)=(x−2)(x2 −x−12)
f left parenthesis x right parenthesis equals left parenthesis x minus 2 right parenthesis left parenthesis x squared minus x minus 12 right parenthesis

f(x)=(x+2)(x+3)(x−4)
f left parenthesis x right parenthesis equals left parenthesis x plus 2 right parenthesis left parenthesis x plus 3 right parenthesis left parenthesis x minus 4 right parenthesis

f(x)=(x−2)(x+3)(x−4)
f left parenthesis x right parenthesis equals left parenthesis x minus 2 right parenthesis left parenthesis x plus 3 right parenthesis left parenthesis x minus 4 right parenthesis

f(x)=(x−2)(x−3)(x−4)

Given that \( x = 2 \) is a zero of the polynomial \( f(x) = x^3 - 3x^2 - 10x + 24 \), we can use the Factor Theorem, which states that \( (x - c) \) is a factor of \( f(x) \) if \( f(c) = 0 \). Since \( x = 2 \) is a zero, \( (x - 2) \) is a factor of \( f(x) \).

Next, we will perform polynomial long division to divide \( f(x) \) by \( (x - 2) \) to determine the other factors.

Step 1: Polynomial Long Division

  1. Divide the leading term: \[ \frac{x^3}{x} = x^2 \]
  2. Multiply: \[ x^2(x - 2) = x^3 - 2x^2 \]
  3. Subtract: \[ (x^3 - 3x^2) - (x^3 - 2x^2) = -x^2 \]
  4. Bring down the next term:
    • Now the polynomial is \( -x^2 - 10x \).
  5. Repeat the process:
    • Divide: \(\frac{-x^2}{x} = -x\)
    • Multiply: \(-x(x - 2) = -x^2 + 2x\)
    • Subtract: \[ (-x^2 - 10x) - (-x^2 + 2x) = -12x \]
  6. Bring down the last term:
    • Now the polynomial is \( -12x + 24 \).
  7. Repeat:
    • Divide: \(\frac{-12x}{x} = -12\)
    • Multiply: \(-12(x - 2) = -12x + 24\)
    • Subtract: \[ (-12x + 24) - (-12x + 24) = 0 \]

Step 2: Summarize the division

Now we can express \( f(x) \) as: \[ f(x) = (x - 2)(x^2 - x - 12) \]

Step 3: Factor the quadratic \( x^2 - x - 12 \)

To factor \( x^2 - x - 12 \), we look for two numbers that multiply to \(-12\) and add to \(-1\). These numbers are \( -4 \) and \( 3 \).

Thus: \[ x^2 - x - 12 = (x - 4)(x + 3) \]

Final Factorization of \( f(x) \)

Putting it all together, we have: \[ f(x) = (x - 2)(x - 4)(x + 3) \]

Conclusion

The final factored form of \( f(x) \) is: \[ f(x) = (x - 2)(x + 3)(x - 4) \]

Therefore, the correct response is: \( f(x) = (x - 2)(x + 3)(x - 4) \).

Use x=1+r
, where r is the interest rate paid each year. Write a model polynomial, C(x)
, that represents the final amount of an 8-year investment if $500 was deposited at the beginning of the first year, $800 was deposited at the beginning of the fourth year, and an additional $370 was deposited at the beginning of the eighth year.(1 point)
Responses

C(x)=500x+800x5+370x8
upper C left parenthesis x right parenthesis equals 500 x plus 800 x superscript 5 baseline plus 370 x superscript 8 baseline

C(x)=500x8+800x4+370x
upper C left parenthesis x right parenthesis equals 500 x superscript 8 baseline plus 800 x superscript 4 baseline plus 370 x

C(x)=500x8+800x4+370
upper C left parenthesis x right parenthesis equals 500 x superscript 8 baseline plus 800 x superscript 4 baseline plus 370

C(x)=500x8+800x5+370x

To model the final amount of an investment made over 8 years with three different deposits, we'll consider the compounding effect of interest for each deposit based on when it was made.

Information:

  1. Initial Deposit: $500 at the beginning of the first year (compounded for 8 years).
  2. Second Deposit: $800 at the beginning of the fourth year (compounded for 5 years).
  3. Third Deposit: $370 at the beginning of the eighth year (compounded for 1 year).

Model Polynomial:

Using \( x = 1 + r \) (where \( r \) is the interest rate), we can express the final amount of each deposit as follows:

  1. For the first deposit of $500: \[ 500 \cdot x^8 \]

  2. For the second deposit of $800 (compounded for 5 years): \[ 800 \cdot x^5 \]

  3. For the third deposit of $370 (compounded for 1 year): \[ 370 \cdot x^1 = 370x \]

Final Model Polynomial:

Putting it all together, the total final amount \( C(x) \) is given by: \[ C(x) = 500x^8 + 800x^5 + 370x \]

Conclusion:

Thus, the correct response is: \( C(x) = 500x^8 + 800x^5 + 370x \).