Polynomial questions

Question

 I have already done it before but am not sure of my answer .

                                          (1)

list the coefficient Identify the given polynomial as monomial,binomial, or trimonial.
3x^3+x^2−8x−13

 

Answer : The Coefficient are 3 ,+1.x^2 , -8 and 13

 

                                         (2)

Rewrite the polynomial in descending order. Identify the leading term,the leading coefficient, and the degree of the polynomial. 13-9x-x^2

 

Answer : -x^2-9x-13

leading term : -x^2

leading coefficent: -1

degree of polynomial : 2

 

                                       (3)

 

Rewrite the polynomial in descending order. Identify the leading term,the leading coefficient, and the degree of the polynomial. 2x^5-x^6+24-3x^3+x-5x^2

 

Answer: x^6-2x^5-3x^3-5x^2+x+24

leading term:x^6

leading coefficient: x

degree of polynomial:6

   

                                       (4)

For the given function f(x) and g(x), find f(x)+g(x) and f(x)-g(x).

 

f(x)=2x^2+11x-5, g(x)=-x^2-5x+20

 

I haven't solved this ,because i don't understand.

 

                                        (5)

For the given function f(x) and g(x), find f(x)+g(x) and f(x)-g(x).

 

f(x)=2x^3-4x^2+8x-16, g(x)=x^3+12x^2-15

 

                                         (6)

Evaluate the polynomial 

3x^2+11xy+y^2 for x=4 and y=3

Answer 1

Section (4)

 

f(x) = 2x² + 11x - 5: g(x) = x² -5x + 20

 

f(x) + g(x) = 2x² + 11x -5 + (x² -5x + 20)

 

                 =x²(2 + 1) + x(11 - 5) + (20 - 5) = 3x² + 6x + 15

 

f(x) - g(x) = 2x² + 11x -5 - (x² -5x + 20) = x²(2 - 1) + x[11 - (-5)] + [20 - (-5)] = x² + 16x - 25

 

Section (5) is analogous to Section (4)

 

f(x) = 2x³ - 4x² + 8x - 16: g(x) = x³ + 12x² - 15

 

f(x) + g(x) = x³ (2 + 1) + x² (-4 + 12) +8x - 16 + (-15) = 3x³ +8x² +8x -31

 

f(x) - g(x) = x³ -16x² +8x -1

 

Section (6)

 

3x² + 11xy +y²  Evaluate at x = 4 and y = 3

 

Substitute 4 for x and 3 for y

 

[3*(4)² + 11*(4)*(3) + (3)² = (3)*(16) + (11)*(12) + 9 = 48 + 132 + 9 = 189

Answer 2

1 is correct. 

2. Almost.  -x^2-9x +13. everything else is right. 

3. Th degree is correct the others have corrections. 

-x^6+2x^5-3x^3-5x^2+x+24

the leading coefficient is the number not the variable. And since it is just x^6 the leading coefficient is 1. 

4. You add (or subtract in the 2nd case) the two functions together and combine like terms.

2x^2+11x-5 +(x^2-5x+20)

=3x^2+6x+15

 

2x^2+11x-5 -(x^2-5x+20). Now you need to distribute the negative. 

2x^2+11x-5 -x^2+5x-20

x^2+16x-25

 

 

Following the the same logic:

5.  f(x)+g(x)=3x^3+8x^2+8x-31

 

f(x)-g(x)=x^3-16x^2+8x-1

 

6. 3x^2+11xy+y^2. X=4 y=3. 

Substitute in the values and remembering the order of operations

3(4^2)+11(4)(3)+(3^2)

3(16)+11(12)+9

48+132+9

189

 

 

 

 

 

Answer 3

1. Very close. Remember that coefficients are the number in front of the x.  So we have 3 (because it's in front of x³), 1 (because it's in front of x²), and -8 (because it's in front of the x). -13 isn't in front of any x so it's just a constant and thus not a coefficient.

    It also asks you to identify the polynomial as a monomial, binomial, or trimonial. This refers to the leading degree, or the highest exponent in the equation. A monomial means that the 1 is the highest degree, binomial means that 2 is the highest degree, trinomial means 3 is the highest degree. In this equation, 3 is the highest exponent => this is a trinomial. 
Answer: Coefficients = 3, 1, -8. This polynomial is a trinomial. 

2. Congrats!! You got this one right all by yourself, and you got every part of this question correct. Serious pat on the back. 

3. Again, you're really really close to the right answer. f(x) = 2x⁵-x⁶+24-3x³+x-5x²

   a. Write equation in descending order. That means we should write the equation so that the degrees are in descending order. You did this mostly right, you got your terms in descending order which is the most important part, you just for got to your negative so since it's -x⁶ the right answer is -x6+2x⁵-3x³-5x²+24.
   b. Leading Term. You got this right as well. It's the first term in an equation that in descending order. In this equation, it's -x^6.
   c. Leading coefficient is the coefficient of the leading term. Since the leading term of this equation is -x⁶ = -1x⁶ => that the leading term is -1
  d. You got this one right. Is your back sore from all the self-congratulations? 


4. f(x) + g(x) just means that you should add the equations together. f(x)=2x²+11x-5, g(x)=-x²-5x+20

               f(x)                  +                    g(x)

      2x² + 11x - 5]           +           [-x² - 5x + 20]
      2x² + 11x - 5 - x² -5x + 20     , drop parenthesis because it's just addition        (2x² -x²) + (11x - 5x) + (-5 + 20)   , add like terms 
                  x²   +       6x      +      15

               f(x)                   -                     g(x)
     [2x² + 11x - 5]            -             [-x² - 5x + 20]
     2x² + 11x - 5 + x² + 5x - 20      , drop parenthesis after distributing the -1 

     (2x² + x²) +(11x + 5x) + (-5 - 20)     , add like terms 

               3x²     +       16x     -      25     

 

 

5. You do this problem exactly like you do problem 4. Try to see if you can do it before looking at the solution. f(x)=2x³-4x²+8x-16, g(x)=x³+12x²-15

f(x) + g(x) = 
[2x³ - 4x² + 8x - 16] + [x³ + 12x² - 15] = 

2x³ - 4x² + 8x - 16 + x³ + 12x² - 15 = 

(2x³ + x³) + (-4x² + 12x²) + 8x +(-16 - 15) =
3x³ + 8x² +8x - 31

f(x) - g(x)

[2x³ - 4x² + 8x - 16] - [x³ + 12x² - 15] =
2x³ - 4x² + 8x - 16 - x³ - 12x² + 15 =
(2x³ - x³) + (-4x² - 12x²) + 8x +(-16 + 15) =

x³ - 16x² + 8x - 1



6. x and y are just variables, they just represent some number. In this problem, it's giving you values for x and y. To solve everywhere you see x you put 4, and everywhere you see y you put 3. 
3x² + 11xy + y² for x=4 and y=3
3(4)² + 11(4)(3) + (3)²           , Order of Operations is imperative to do this problem correctly. PEMDAS
3(16) + 11(4)(3) + 9
48 + 132 + 9

189