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777dan777 [17]
3 years ago
15

1. Suppose f(x) = x^4-2x^3+ax^2+x+3. If f(3) = 2, then what is a? 2. Let f, g, and h be polynomials such that h(x) = f(x) * g(x)

. If the constant term of f(x) is -4 and the constant term of h(x) is 3, what is g(0)? 3. Suppose the polynomials f and g are both monic polynomials. If the sum f(x) + g(x) is also monic, what can we deduce about the degrees of f and g? 4. If f(x) is a polynomial, is f(x^2) also a polynomial 5. Consider the polynomial function g(x) = x^4-3x^2+9 a. What must be true of a polynomial function f(x) if f(x) and f(-x) are the same polynomial b.What must be true of a polynomial function f(x) if f(x) and -f(-x) are the same polynomial
Mathematics
1 answer:
vagabundo [1.1K]3 years ago
7 0

Answer:

1. a = -31/9

2. -3/4

3. Different degree polynomials

4. Yes, of a degree 2n

5. a. Even-degree variables

b. Odd- degree variables

Step-by-step explanation:

1. Suppose f(x) = x^4-2x^3+ax^2+x+3. If f(3) = 2, then what is a?

<u>Plugging in 3 for x:</u>

f(3)= 3^4 - 2*3^3 + a*3^2 + 3 + 3= 81 - 54 + 6 + 9a = 33 + 9a and f(3)= 2

  • 9a+33= 2
  • 9a= -31
  • a = -31/9

------------

2. Let f, g, and h be polynomials such that h(x) = f(x) * g(x). If the constant term of f(x) is -4 and the constant term of h(x) is 3, what is g(0)?

  • f(0)= -4, h(0)= 3, g(0) = ?
  • h(x)= f(x)*g(x)
  • g(x)= h(x)/f(x)
  • g(0) = h(0)/f(0) = 3/-4= -3/4
  • g(0)= -3/4

------------

3. Suppose the polynomials f and g are both monic polynomials. If the sum f(x) + g(x) is also monic, what can we deduce about the degrees of f and g?

  • <em>A monic polynomial is a single-variable polynomial in which the leading coefficient is equal to 1. </em>

If the sum of monic polynomials f(x) + g(x) is also monic, then f(x) and g(x) are of different degree and their sum only change the one with the lower degree, leaving the higher degree variable unchanged.

------------

4. If f(x) is a polynomial, is f(x^2) also a polynomial?

  • If f(x) is a polynomial of degree n, then f(x^2) is a polynomial of degree 2n

------------

5. Consider the polynomial function g(x) = x^4-3x^2+9

a. What must be true of a polynomial function f(x) if f(x) and f(-x) are the same polynomial?

  • If f(x) and f(-x) are same polynomials, then they have even-degree variables.

b.What must be true of a polynomial function f(x) if f(x) and -f(-x) are the same polynomial?

  • If f(x) and -f(-x) are the same polynomials, then they have odd-degree variables.
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A rectangular swimming pool is twice as long as it is wide. A small concrete walkway surrounds the pool. The walkway is a 2 feet
Ksivusya [100]

Answer:

The width and the length of the pool are 12 ft and 24 ft respectively.

Step-by-step explanation:

The length (L) of the rectangular swimming pool is twice its wide (W):

L_{1} = 2W_{1}

Also, the area of the walkway of 2 feet wide is 448:

W_{2} = 2 ft

A_{T} = W_{2}*L_{2} = 448 ft^{2}

Where 1 is for the swimming pool (lower rectangle) and 2 is for the walkway more the pool (bigger rectangle).

The total area is related to the pool area and the walkway area as follows:

A_{T} = A_{1} + A_{w}    (1)          

The area of the pool is given by:

A_{1} = L_{1}*W_{1}        

A_{1} = (2W_{1})*W_{1} = 2W_{1}^{2}  (2)          

And the area of the walkway is:

A_{w} = 2(L_{2}*2 + W_{1}*2) = 4L_{2} + 4W_{1}    (3)          

Where the length of the bigger rectangle is related to the lower rectangle as follows:                  

L_{2} = 4 + L_{1} = 4 + 2W_{1}   (4)        

By entering equations (4), (3), and (2) into equation (1) we have:

A_{T} = A_{1} + A_{w}

A_{T} = 2W_{1}^{2} + 4L_{2} + 4W_{1}                

448 = 2W_{1}^{2} + 4(4 + 2W_{1}) + 4W_{1}            

224 = W_{1}^{2} + 8 + 4W_{1} + 2W_{1}

224 = W_{1}^{2} + 8 + 6W_{1}

By solving the above quadratic equation we have:

W₁ = 12 ft

Hence, the width of the pool is 12 feet, and the length is:

L_{1} = 2W_{1} = 2*12 ft = 24 ft

Therefore, the width and the length of the pool are 12 ft and 24 ft respectively.

I hope it helps you!                                                                                          

8 0
3 years ago
HELP PLEASE <br> solve the right triangle
aleksandrvk [35]

Answer:

Part 1) FG=4.4\ units

Part 2) EF=3.9\ units

Part 3) m\angle G=63^o

Step-by-step explanation:

step 1

Find the measure of length side FG

In the right triangle EFG

we know that

sin(27^o)=\frac{GE}{FG} ----> by SOH (opposite side divided by the hypotenuse)

substitute the given values

sin(27^o)=\frac{2}{FG}

FG=\frac{2}{sin(27^o)}=4.4\ units

step 2

Find the measure of length side EF

In the right triangle EFG

we know that

cos(27^o)=\frac{EF}{FG} ----> by CAH (adjacent side divided by the hypotenuse)

substitute the given values

cos(27^o)=\frac{EF}{4.4}

EF=cos(27^o)(4.4)=3.9\ units

step 3

Find the measure of angle G

we know that

m\angle G+27^o=90^o ---> by complementary angles in a right triangle

m\angle G=90^o-27^o=63^o

7 0
3 years ago
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