perform necessary operations. (14t^4-4t^2+8t)+(4t^4-8t+27)-(8t^4+t^2+14)
To perform the given operations, we need to combine like terms in the expression:
(14t^4 - 4t^2 + 8t) + (4t^4 - 8t + 27) - (8t^4 + t^2 + 14)
The first step is to combine like terms within each of the parentheses:
(14t^4 + 4t^4 - 8t^4) + (-4t^2 - t^2) + (8t - 8t) + 27 - 14
Simplifying further, we get:
10t^4 - 5t^2 + 0t + 27 - 14
Finally, combining like terms by adding or subtracting, we have:
10t^4 - 5t^2 + 13
Subtract the polynomials.
(4t^3 - 8t + 19) - (-16t^3 + t^2 + 11t)
To subtract the polynomials, we need to distribute the negative sign to each term within the second polynomial:
(4t^3 - 8t + 19) - (-16t^3 + t^2 + 11t)
= 4t^3 - 8t + 19 + 16t^3 - t^2 - 11t
Next, we can rearrange the terms in descending order of the degree:
= (4t^3 + 16t^3) + (- t^2 - 8t - 11t) + 19
= 20t^3 - 20t - t^2 + 19
Thus, the subtraction of the polynomials is 20t^3 - 20t - t^2 + 19.
Add the polynomials.
(-10t^5 + 9t^4 - 3t^3 + 2t + 11) + (-6 - 3t - 3t^2 + 3t^3 + 3t^5)
To add the polynomials, we need to combine like terms:
(-10t^5 + 9t^4 - 3t^3 + 2t + 11) + (-6 - 3t - 3t^2 + 3t^3 + 3t^5)
Reordering the terms:
-10t^5 + 3t^5 + 9t^4 - 3t^3 + 3t^3 - 3t^2 - 3t + 2 + 11 - 6
Combining like terms:
(-10t^5 + 3t^5) + 9t^4 + 0t^3 - 3t^2 - 3t + 2 + 11 - 6
Simplifying further:
-7t^5 + 9t^4 - 3t^2 - 3t + 7
Thus, the sum of the polynomials is -7t^5 + 9t^4 - 3t^2 - 3t + 7.
Multiply the polynomials.
-2y^5(6y - y^2)
To multiply the polynomials, we will use the distributive property.
-2y^5(6y - y^2)
First, distribute -2y^5 to each term within the parentheses:
-2y^5 * 6y + -2y^5 * -y^2
Next, simplify each term by multiplying:
-12y^6 + 2y^7
Thus, the product of the polynomials is -12y^6 + 2y^7.
Multiply the polynomials.
(y^2 + y - 3)(2y^2 + 2y - 1)
To multiply the polynomials, we can use the distributive property and multiply each term in the first polynomial by each term in the second polynomial:
(y^2 + y - 3)(2y^2 + 2y - 1)
Multiplying each term, we get:
y^2 * 2y^2 + y^2 * 2y - y^2 * 1 + y * 2y^2 + y * 2y - y * 1 - 3 * 2y^2 - 3 * 2y + 3 * 1
Simplifying each term, we have:
2y^4 + 2y^3 - y^2 + 2y^3 + 2y^2 - y - 6y^2 - 6y + 3
Now, combining like terms, we get:
2y^4 + 4y^3 - 5y^2 - 7y + 3
Thus, the product of the polynomials is 2y^4 + 4y^3 - 5y^2 - 7y + 3.