Multiply the polynomial expression (x+2)(x−4).
To multiply the polynomial expression (x+2)(x-4), we can use the distributive property.
We start by multiplying the first terms of each binomial:
(x)(x) = x^2
Then, we multiply the outer terms:
(x)(-4) = -4x
Next, we multiply the inner terms:
(2)(x) = 2x
Finally, we multiply the last terms of each binomial:
(2)(-4) = -8
Putting it all together, we have:
(x+2)(x-4) = x^2 - 4x + 2x - 8
Combining like terms, we have:
x^2 - 2x - 8
So, the final answer is x^2 - 2x - 8.
Multiply to simplify the polynomial expression (r−1)*(r^2−2r+3).
To multiply and simplify the polynomial expression (r−1)(r^2−2r+3), we will use the distributive property.
First, we multiply the first term of the first binomial (r) by each term of the second binomial (r^2−2r+3):
(r)(r^2) = r^3
(r)(-2r) = -2r^2
(r)(3) = 3r
Next, we multiply the second term of the first binomial (-1) by each term of the second binomial:
(-1)(r^2) = -r^2
(-1)(-2r) = 2r
(-1)(3) = -3
Now, we add up all the resulting terms:
r^3 - 2r^2 + 3r - r^2 + 2r - 3
Combining like terms, we have:
r^3 - 3r^2 + 5r - 3
So, the simplified polynomial expression is r^3 - 3r^2 + 5r - 3.
Simplify the polynomial expression (xy−2)(x^2+1).
To simplify the polynomial expression (xy−2)(x^2+1), we will again use the distributive property.
First, we multiply the first term of the first binomial (xy) by each term of the second binomial (x^2+1):
(xy)(x^2) = x^3y
(xy)(1) = xy
Next, we multiply the second term of the first binomial (-2) by each term of the second binomial:
(-2)(x^2) = -2x^2
(-2)(1) = -2
Now, we add up all the resulting terms:
x^3y + xy -2x^2 - 2
Combining like terms, we have:
x^3y - 2x^2 + xy - 2
So, the simplified polynomial expression is x^3y - 2x^2 + xy - 2.
Which of the following shows that polynomials form a closed system under multiplication?
Option #1: 3(2x+5)=6x+15
Option #2: 8x+7=5x+3x+1+6
Option #3: (2x2−4)(3y+6)=6x2y+12x2−12y−24
Option #4: 2x12⋅3=6x−−√
The correct option that shows that polynomials form a closed system under multiplication is Option #3: (2x^2−4)(3y+6)=6x^2y+12x^2−12y−24.
This option demonstrates the multiplication of two polynomials and shows that the result is also a polynomial.
Which of the following options correctly describes if the following polynomial forms a closed system under multiplication?−3(5x+4y)=−15x−12y
Option #1: Yes, because the result of multiplying the polynomials is also a polynomial.
Option #2: No, because the exponent of x is not a positive integer.
Option #3: No, because multiplying the polynomials resulted in subtraction.
Option #1: Yes, because the result of multiplying the polynomials is also a polynomial.
In the given expression, we have −3(5x+4y)=−15x−12y. When we multiply the polynomial −3 by the polynomial (5x+4y), we get −15x−12y, which is a polynomial as well. Therefore, this expression demonstrates that the polynomial forms a closed system under multiplication.