Simone used a CAS to multiply 2n-5 by n^2+3n+6 to get 2*n^3+n^2-3*n-30
explain which multiplications were done and combined to get each term in the product
Simone used a CAS to multiply 2n-5 by n^2+3n+6 to get 2*n^3+n^2-3*n-30
explain which multiplications were done and combined to get each term in the product
step 1: 2n-5 (multiply) n^2+3n+6
step 2: * Multiply 2n to each.. (n^2+3n+6)
2n * n^2 = 2n^3; 2n * 3n = 6n^2; 2n * 6 = 12n; therefore (2n^3+6n^2+12n) keep this for step#4
step 3: * Multiply -5 to each.. (n^2+3n+6)
-5 * n^2 = -5n^2; -5 * 3n = -15n; -5 * 6 = -30; therefore (-5n^2 -15n -30)
Step 4: Combine (2n^3+6n^2+12n) & (-5n^2 -15n -30) and put them in order
2n^3 = complete; 6n^2 - 5n^2 = n^2; 12n - 15n = -3n; -30 = complete.
Step 5: Combine all answers together and you have = 2n^3 +n^2 - 3n - 30
I hope the steps were clear and you understood! =)
When multiplying polynomials such as (3x + 7) * (x^2 - 3x + y), you multiply every expression in the first one by every expression in the second one, so you end up with 3x*x^{2} + 3x*(-)3y + 3x*y + 7*x^{2} + 7*(-)3x + 7*y. The (-) means "negative" and not "subtract" here.
When multiplying exponents, like x^{3}*x^{5}, you keep the x the same, and add the exponent numbers, so in this case it would be x^{3+5 }or x^{8}. Also, when multiplying a negative number and a positive number, you get a negative number. So simplifying each result of the multiplication in the example you would get (bold expressions are the simplified forms):
3x*x^{2} => 3x^{1+2} => 3x^{3}
3x*(-)3y => 3*(-)3*x*y => (-)9xy
3x*y => 3xy
7*x^{2} => 7x^{2}
7*(-)3x => (-)21x
7*y => 7y
Now that you know what each expression is, you want to try to group the like expressions together so you can add/subtract more easily (you go by the variables and their exponents so 2xy and 4xy would be like expressions but 2xy^{2} and 4x^{2}y wouldn't be like expressions). So piecing the simplified expressions together so the like expressions are close you would get:
3x^{3} + 7x^{2} + 3xy - 9xy - 21x + 7y (Actual order is a matter of preference, I find it easier to start with the lowest letter variable and then from there go with the highest exponent of that letter, then do the same with the second, third, or even fourth variable, going up by letters and down by exponents.)
Now that you have like variables near each other, you can perform addition/subtraction on them.
So you end up with 3x^{3} + 7x^{2} + (3-9)xy -21x + 7y or, simplified, 3x^{3} + 7x^{2} - 6xy - 21x + 7y