Multiply the polynomials:
1. (x^2+3x-1)(2x^2-5x-1)
2. (2x-1)(x^4-3x^3+x2-5x+12)
there are lots of polynomial multiplication web sites. They will show you the workings if you have trouble. The first one is
2x^4 + x^3 - 18x^2 + 2x + 1
You try the other, and show us what happens if you get stuck.
Multiply polynomials just like multi-digit numbers, keeping each power of x in its own column, and no need to carry between columns.
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To multiply polynomial expressions, we use the distributive property and combine like terms. Let's tackle each problem step by step:
1. (x^2 + 3x - 1)(2x^2 - 5x - 1)
To multiply these polynomials, we need to multiply each term in the first polynomial by every term in the second polynomial and then combine like terms.
We start by multiplying the first term of the first polynomial (x^2) by each term in the second polynomial:
x^2 * 2x^2 = 2x^4
x^2 * -5x = -5x^3
x^2 * -1 = -x^2
Next, we multiply the second term of the first polynomial (3x) by each term in the second polynomial:
3x * 2x^2 = 6x^3
3x * -5x = -15x^2
3x * -1 = -3x
Finally, we multiply the third term of the first polynomial (-1) by each term in the second polynomial:
-1 * 2x^2 = -2x^2
-1 * -5x = 5x
-1 * -1 = 1
Now, we combine all the terms we obtained:
2x^4 - 5x^3 - x^2 + 6x^3 - 15x^2 - 3x - 2x^2 + 5x + 1
Combining like terms, we get:
2x^4 + (6x^3 - 5x^3) + (-15x^2 - 2x^2 - x^2) + (5x - 3x) + 1
2x^4 + x^3 - 18x^2 + 2x + 1
Therefore, the product of (x^2 + 3x - 1)(2x^2 - 5x - 1) is 2x^4 + x^3 - 18x^2 + 2x + 1.
2. (2x - 1)(x^4 - 3x^3 + x^2 - 5x + 12)
We'll follow the same steps as before.
Multiply the first term of the first polynomial (2x) by each term in the second polynomial:
(2x)(x^4) = 2x^5
(2x)(-3x^3) = -6x^4
(2x)(x^2) = 2x^3
(2x)(-5x) = -10x^2
(2x)(12) = 24x
Next, multiply the second term of the first polynomial (-1) by each term in the second polynomial:
(-1)(x^4) = -x^4
(-1)(-3x^3) = 3x^3
(-1)(x^2) = -x^2
(-1)(-5x) = 5x
(-1)(12) = -12
Combine all the terms obtained:
2x^5 - 6x^4 + 2x^3 - 10x^2 + 24x - x^4 + 3x^3 - x^2 + 5x - 12
Combining like terms, we get:
2x^5 + (-6x^4 - x^4) + (2x^3 + 3x^3) + (-10x^2 - x^2) + (24x + 5x) - 12
2x^5 - 7x^4 + 5x^3 - 11x^2 + 29x - 12
Therefore, the product of (2x - 1)(x^4 - 3x^3 + x^2 - 5x + 12) is 2x^5 - 7x^4 + 5x^3 - 11x^2 + 29x - 12.