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Ganezh [65]
3 years ago
12

Write a polynomial function with the given zeros. multiply the terms (write in standard form).

Mathematics
2 answers:
ryzh [129]3 years ago
7 0

1) \ f(x)=x^3-3x-2 \\ \\ 2) \ g(x)=x^3-x^2-6x

<h2>Explanation:</h2>

A \ \mathbf{polynomial \ function} \ of \ x \ with \ degree \ n \ is \ given \ by:\\ \\ f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots +a_{2}x^{2}+a_{1}x+a_{0} \\ \\ where \ n \ is \ a \ nonnegative \ integer \ and \ a_{n}, a_{n-1}, \ldots a_{2}, a_{1}, a_{0} \\ with \ a_{n}\neq 0

In this exercise, we have two cases and both will be polynomial functions with degree 3 because we have three real roots in each case. So:

<h3>First.</h3>

The roots are:

x= -1, -1, 2

So we can write this polynomial functions as the product of linear factors:

f(x)=(x-(-1))(x-(-1))(x-2) \\ \\ f(x)=(x+1)^2(x-2)

Since we have to write it in standard form, let's expand:

f(x)=(x+1)^2(x-2) \\ \\ \\ By \ property: \\ \\ (a+b)^2=a^2+2ab+b^2 \\ \\ \\ f(x)=(x^2+2x+1)(x-2) \\ \\ f(x)=x^3+2x^2+x-2x^2-4x-2 \\ \\ \\ Combine \ like \ terms: \\ \\ x^3+(2x^2-2x^2)+(x-4x)-2 \\ \\ \\ \boxed{f(x)=x^3-3x-2}

<h3>Second.</h3>

The roots are:

x= -2, 0, 3

Writing the polynomial function as the product of linear factors:

g(x)=(x-(-2))(x-0)(x-3) \\ \\ g(x)=(x+2)x(x-3) \\ \\ g(x)=x(x+2)(x-3) \\ \\ \\ Distributive \property: \\ \\ g(x)=x(x^2-3x+2x-6) \\ \\ g(x)=x(x^2-x-6) \\ \\ \\ Distributive \ property \ again: \\ \\ \boxed{g(x)=x^3-x^2-6x}

<h2>Learn more:</h2>

Degree of polynomial functions: brainly.com/question/5451252

#LearnWithBrainly

zhannawk [14.2K]3 years ago
5 0

1) polynomial having zeros x=-1,-1,2 is x^3-3x-2=0

2) polynomial having zeros x= -2, 0, 3 is x^3-x^2-6x=0\\

Step-by-step explanation:

1) We are given the zeros:

x=-1,-1,2

we need to find the polynomial

x=-1, x=-1, x=2

x+1=0,x+1=0, x-2=0

Multiplying all terms:

(x+1)(x+1)(x-2)=0\\(x(x+1)+1(x+1))(x-2)=0\\(x^2+x+x+1)(x-2)=0\\(x^2+2x+1)(x-2)=0\\x(x^2+2x+1)-2(x^2+2x+1)=0\\x^3+2x^2+x-2x^2-4x-2=0\\x^3+0x^2-3x-2=0\\x^3-3x-2=0

So, polynomial having zeros x=-1,-1,2 is x^3-3x-2=0

2) We are given the zeros:

x= -2, 0, 3

we need to find the polynomial

x= -2, x=0, x=3

x+2=0,x=0, x-3=0

Multiplying all terms:

x(x+2)(x-3)=0\\x(x(x-3)+2(x-3))=0\\x(x^2-3x+2x-6)=0\\x(x^2-x-6)=0\\x^3-x^2-6x=0\\

So, polynomial having zeros x= -2, 0, 3 is x^3-x^2-6x=0\\

Keywords: Zeros of polynomials

Learn more about zeros of polynomials at:

  • brainly.com/question/1414350
  • brainly.com/question/2568692
  • brainly.com/question/1332667
  • brainly.com/question/1357167

#learnwithBrainly

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<h3>Answer:     t = 7</h3>

===========================================================

Explanation:

If we replace every copy of x with t, then we go from this

F(x) = 3x+5

to this

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All we've done really is change the letter. We still are dealing with a variable. We're told that F(t) is equal to 26, which would allow us to replace the "F(t)" with "26" in the second equation above.

So we now have the equation 26 = 3t+5 which is the same as 3t+5 = 26

Let's solve for t

3t+5 = 26

3t = 26-5 .... subtract 5 from both sides

3t = 21

t = 21/3 .... divide both sides by 3

t = 7  is the answer

Now note that...

F(t) = 3t + 5

F(7) = 3*7 + 5 .... replace t with 7

F(7) = 21 + 5

F(7) = 26

This means we got F(t) = 26 when t = 7

It's the same as saying x = 7 leads to F(x) = 26.

This helps confirm we have the correct answer.

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