Find the product of (x^2 - 3x + 5) with the quotient of (10x^6 - 15x^5 - 5x^3) ÷ 5x^3.
I need some help with this. I'm really unclear on the steps and how to solve this as a whole. I don't just need an answer, I'd like to know the steps as well. Please?
Nevermind, I figured it out:
The quotient:
(10x^6 - 15x^5 - 5x^3)/(5x^3 )
(10x^6)/(5x^3 ) - (15x^5)/(5x^3 ) - (5x^3)/(5x^3 )
2x3 - (15x^5)/(5x^3 ) - (5x^3)/(5x^3 )
2x3 - 3x2 - (5x^3)/(5x^3 )
2x3 - 3x2 - 1
The product using the quotient:
(x2 - 3x + 5)(2x3 - 3x2 - 1)
x2 ∙ 2x3 + x2 ∙ -3x2 + x2 ∙ - 1 - 3x ∙ 2x3 - 3x ∙ -3x2 - 3x ∙ -1 + 5 ∙ 2x3 + 5 ∙ -3x2 + 5 ∙ -1
2x5 - 9x4 + 19x3 - 16x2 + 3x - 5
Here's a better format. Good luck to anyone else who has this problem.
The quotient:
(10x^6 - 15x^5 - 5x^3)/(5x^3 )
(10x^6)/(5x^3 ) - (15x^5)/(5x^3 ) - (5x^3)/(5x^3 )
2x^3 - (15x^5)/(5x^3 ) - (5x^3)/(5x^3 )
2x^3 - 3x^2 - (5x^3)/(5x^3 )
2x^3 - 3x^2 - 1
The product using the quotient:
(x^2 - 3x + 5)(2x^3 - 3x^2 - 1)
x^2 * 2x^3 + x^2 * -3x^2 + x^2 * - 1 - 3x * 2x^3 - 3x * -3x^2 - 3x * -1 + 5 * 2x^3 + 5 * -3x^2 + 5 * -1
2x^5 - 9x^4 + 19x^3 - 16x^2 + 3x - 5
The idea is to solve the quotient first, by dividing each term by 5x^3.
Once the solution is found: 2x^3 - 3x^2 - 1
Then use that quotient and multiply it by (x^2 - 3x + 5).
Thus: (x^2 - 3x + 5)(2x^3 - 3x^2 - 1)
Of course! I'll guide you through the steps to solve this problem.
Step 1: Simplify the quotient expression.
To simplify the quotient expression (10x^6 - 15x^5 - 5x^3) ÷ 5x^3, divide each term by 5x^3.
So, you will have (10x^6 ÷ 5x^3) - (15x^5 ÷ 5x^3) - (5x^3 ÷ 5x^3).
Step 2: Simplify each term.
Simplifying each term, you get:
- 10x^(6-3)
- 15x^(5-3)
- 5x^(3-3)
Simplifying the exponents, you will have:
- 10x^3
- 15x^2
- 5
So, the simplified quotient expression is 10x^3 - 15x^2 - 5.
Step 3: Multiply the simplified quotient with the polynomial.
Now, you need to multiply (x^2 - 3x + 5) with the simplified quotient 10x^3 - 15x^2 - 5.
To do this, distribute each term of the simplified quotient to every term in the polynomial and then combine like terms.
(x^2 - 3x + 5) * (10x^3 - 15x^2 - 5)
= (x^2 * 10x^3 - x^2 * 15x^2 - x^2 * 5) + (-3x * 10x^3 - (-3x) * 15x^2 - (-3x) * 5) + (5 * 10x^3 - 5 * 15x^2 - 5 * 5)
Simplifying each term further:
= 10x^5 - 15x^4 - 5x^2 - 30x^4 + 45x^3 + 15x - 50x^3 + 75x^2 + 25
Combine like terms:
= 10x^5 - (15+30)x^4 + (45-50)x^3 + (15+75)x^2 + 15x + 25
= 10x^5 - 45x^4 - 5x^3 + 90x^2 + 15x + 25
So, the product of (x^2 - 3x + 5) with the quotient of (10x^6 - 15x^5 - 5x^3) ÷ 5x^3 is 10x^5 - 45x^4 - 5x^3 + 90x^2 + 15x + 25.