Descartes' Rule, Intermediate Value Theorem, Sum and Product of Zeros - Pre-Calculus
Card 1 of 16
Please choose the best answer from the following choices.
Find the sum and product of the zeros of the following polynomial:
Please choose the best answer from the following choices.
Find the sum and product of the zeros of the following polynomial:
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Factoring the polynomial will give you 
and 
.
The zeros of these binomials will be 
and 
.
If you add these together (sum) you get 
and if you multiply them together (product) you get 
.
Factoring the polynomial will give you
The zeros of these binomials will be
If you add these together (sum) you get
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Determine the possible number of Positive and Negative Real Zeros of the polynomial using Descartes' Rule of Signs
Determine the possible number of Positive and Negative Real Zeros of the polynomial using Descartes' Rule of Signs
Tap to reveal answer
In order to determine the positive number of real zeroes, we must count the number of sign changes in the coefficients of the terms of the polynomial. The number of real zeroes can then be any positive difference of that number and a positive multiple of two.
For the function
there are four sign changes. As such, the number of positive real zeroes can be
In order to determine the positive number of real zeroes, we must count the number of sign changes in the coefficients of the terms of the polynomial after substituting for 
The number of real zeroes can then be any positive difference of that number and a positive multiple of two.
After substituting 
we get
and there is one sign change.
As such, there can only be one negative root.
In order to determine the positive number of real zeroes, we must count the number of sign changes in the coefficients of the terms of the polynomial. The number of real zeroes can then be any positive difference of that number and a positive multiple of two.
For the function
there are four sign changes. As such, the number of positive real zeroes can be
In order to determine the positive number of real zeroes, we must count the number of sign changes in the coefficients of the terms of the polynomial after substituting for
After substituting
and there is one sign change.
As such, there can only be one negative root.
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Given 
, determine the sum and product of the zeros respectively.
Given
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To determine zeros of 
, factorize the polynomial.
Set each of the factorized components equal to zero and solve for 
.
The sum of the roots:
The product of the roots:
To determine zeros of
Set each of the factorized components equal to zero and solve for
The sum of the roots:
The product of the roots:
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Please choose the best answer from the following choices.
Find the sum and product of the zeros of the following polynomial:
Please choose the best answer from the following choices.
Find the sum and product of the zeros of the following polynomial:
Tap to reveal answer
To find the zeros you have to factor the polynomial.
This is easily factorable and you will get 
and 
.
Next, set both of these equal to zero. 
and 
.
Isolate the x's and you will get 
and 
.
The sum will be 
since you add the two together, and the product will be 
because you multiply the two together.
To find the zeros you have to factor the polynomial.
This is easily factorable and you will get
Next, set both of these equal to zero.
Isolate the x's and you will get
The sum will be
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Please choose the best answer from the following choices.
Find the sum and product of the zeros of the following polynomial:
Please choose the best answer from the following choices.
Find the sum and product of the zeros of the following polynomial:
Tap to reveal answer
Factoring the polynomial will give you and .
The zeros of these binomials will be and .
If you add these together (sum) you get and if you multiply them together (product) you get .
Factoring the polynomial will give you
The zeros of these binomials will be
If you add these together (sum) you get
← Didn't Know|Knew It →
Determine the possible number of Positive and Negative Real Zeros of the polynomial using Descartes' Rule of Signs
Determine the possible number of Positive and Negative Real Zeros of the polynomial using Descartes' Rule of Signs
Tap to reveal answer
In order to determine the positive number of real zeroes, we must count the number of sign changes in the coefficients of the terms of the polynomial. The number of real zeroes can then be any positive difference of that number and a positive multiple of two.
For the function
there are four sign changes. As such, the number of positive real zeroes can be
In order to determine the positive number of real zeroes, we must count the number of sign changes in the coefficients of the terms of the polynomial after substituting for The number of real zeroes can then be any positive difference of that number and a positive multiple of two.
After substituting we get
and there is one sign change.
As such, there can only be one negative root.
In order to determine the positive number of real zeroes, we must count the number of sign changes in the coefficients of the terms of the polynomial. The number of real zeroes can then be any positive difference of that number and a positive multiple of two.
For the function
there are four sign changes. As such, the number of positive real zeroes can be
In order to determine the positive number of real zeroes, we must count the number of sign changes in the coefficients of the terms of the polynomial after substituting for
After substituting
and there is one sign change.
As such, there can only be one negative root.
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Given , determine the sum and product of the zeros respectively.
Given
Tap to reveal answer
To determine zeros of , factorize the polynomial.
Set each of the factorized components equal to zero and solve for .
The sum of the roots:
The product of the roots:
To determine zeros of
Set each of the factorized components equal to zero and solve for
The sum of the roots:
The product of the roots:
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Please choose the best answer from the following choices.
Find the sum and product of the zeros of the following polynomial:
Please choose the best answer from the following choices.
Find the sum and product of the zeros of the following polynomial:
Tap to reveal answer
To find the zeros you have to factor the polynomial.
This is easily factorable and you will get and .
Next, set both of these equal to zero. and .
Isolate the x's and you will get and .
The sum will be since you add the two together, and the product will be because you multiply the two together.
To find the zeros you have to factor the polynomial.
This is easily factorable and you will get
Next, set both of these equal to zero.
Isolate the x's and you will get
The sum will be
← Didn't Know|Knew It →
Determine the possible number of Positive and Negative Real Zeros of the polynomial using Descartes' Rule of Signs
Determine the possible number of Positive and Negative Real Zeros of the polynomial using Descartes' Rule of Signs
Tap to reveal answer
In order to determine the positive number of real zeroes, we must count the number of sign changes in the coefficients of the terms of the polynomial. The number of real zeroes can then be any positive difference of that number and a positive multiple of two.
For the function
there are four sign changes. As such, the number of positive real zeroes can be
In order to determine the positive number of real zeroes, we must count the number of sign changes in the coefficients of the terms of the polynomial after substituting for The number of real zeroes can then be any positive difference of that number and a positive multiple of two.
After substituting we get
and there is one sign change.
As such, there can only be one negative root.
In order to determine the positive number of real zeroes, we must count the number of sign changes in the coefficients of the terms of the polynomial. The number of real zeroes can then be any positive difference of that number and a positive multiple of two.
For the function
there are four sign changes. As such, the number of positive real zeroes can be
In order to determine the positive number of real zeroes, we must count the number of sign changes in the coefficients of the terms of the polynomial after substituting for
After substituting
and there is one sign change.
As such, there can only be one negative root.
← Didn't Know|Knew It →
Given , determine the sum and product of the zeros respectively.
Given
Tap to reveal answer
To determine zeros of , factorize the polynomial.
Set each of the factorized components equal to zero and solve for .
The sum of the roots:
The product of the roots:
To determine zeros of
Set each of the factorized components equal to zero and solve for
The sum of the roots:
The product of the roots:
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Please choose the best answer from the following choices.
Find the sum and product of the zeros of the following polynomial:
Please choose the best answer from the following choices.
Find the sum and product of the zeros of the following polynomial:
Tap to reveal answer
To find the zeros you have to factor the polynomial.
This is easily factorable and you will get and .
Next, set both of these equal to zero. and .
Isolate the x's and you will get and .
The sum will be since you add the two together, and the product will be because you multiply the two together.
To find the zeros you have to factor the polynomial.
This is easily factorable and you will get
Next, set both of these equal to zero.
Isolate the x's and you will get
The sum will be
← Didn't Know|Knew It →
Please choose the best answer from the following choices.
Find the sum and product of the zeros of the following polynomial:
Please choose the best answer from the following choices.
Find the sum and product of the zeros of the following polynomial:
Tap to reveal answer
Factoring the polynomial will give you and .
The zeros of these binomials will be and .
If you add these together (sum) you get and if you multiply them together (product) you get .
Factoring the polynomial will give you
The zeros of these binomials will be
If you add these together (sum) you get
← Didn't Know|Knew It →
Determine the possible number of Positive and Negative Real Zeros of the polynomial using Descartes' Rule of Signs
Determine the possible number of Positive and Negative Real Zeros of the polynomial using Descartes' Rule of Signs
Tap to reveal answer
In order to determine the positive number of real zeroes, we must count the number of sign changes in the coefficients of the terms of the polynomial. The number of real zeroes can then be any positive difference of that number and a positive multiple of two.
For the function
there are four sign changes. As such, the number of positive real zeroes can be
In order to determine the positive number of real zeroes, we must count the number of sign changes in the coefficients of the terms of the polynomial after substituting for The number of real zeroes can then be any positive difference of that number and a positive multiple of two.
After substituting we get
and there is one sign change.
As such, there can only be one negative root.
In order to determine the positive number of real zeroes, we must count the number of sign changes in the coefficients of the terms of the polynomial. The number of real zeroes can then be any positive difference of that number and a positive multiple of two.
For the function
there are four sign changes. As such, the number of positive real zeroes can be
In order to determine the positive number of real zeroes, we must count the number of sign changes in the coefficients of the terms of the polynomial after substituting for
After substituting
and there is one sign change.
As such, there can only be one negative root.
← Didn't Know|Knew It →
Given , determine the sum and product of the zeros respectively.
Given
Tap to reveal answer
To determine zeros of , factorize the polynomial.
Set each of the factorized components equal to zero and solve for .
The sum of the roots:
The product of the roots:
To determine zeros of
Set each of the factorized components equal to zero and solve for
The sum of the roots:
The product of the roots:
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Please choose the best answer from the following choices.
Find the sum and product of the zeros of the following polynomial:
Please choose the best answer from the following choices.
Find the sum and product of the zeros of the following polynomial:
Tap to reveal answer
To find the zeros you have to factor the polynomial.
This is easily factorable and you will get and .
Next, set both of these equal to zero. and .
Isolate the x's and you will get and .
The sum will be since you add the two together, and the product will be because you multiply the two together.
To find the zeros you have to factor the polynomial.
This is easily factorable and you will get
Next, set both of these equal to zero.
Isolate the x's and you will get
The sum will be
← Didn't Know|Knew It →
Please choose the best answer from the following choices.
Find the sum and product of the zeros of the following polynomial:
Please choose the best answer from the following choices.
Find the sum and product of the zeros of the following polynomial:
Tap to reveal answer
Factoring the polynomial will give you and .
The zeros of these binomials will be and .
If you add these together (sum) you get and if you multiply them together (product) you get .
Factoring the polynomial will give you
The zeros of these binomials will be
If you add these together (sum) you get
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