Can someone show me step by step how to get the sum diff and product of this problem x^3- 〖2x〗^2 + 3x-1 ∶ x^2- x+4
Sure! To find the sum, difference, and product of the two expressions, we will need to perform a series of mathematical operations. Here are the step-by-step instructions:
Step 1: Simplify both expressions separately.
Let's simplify the expressions:
Expression 1: x^3 - 2x^2 + 3x - 1
Expression 2: x^2 - x + 4
Step 2: Find the sum (addition) of the two expressions.
To add the expressions, simply add the corresponding coefficients of the same degrees of x. In other words, add the coefficients of the x^3 terms, the x^2 terms, the x terms, and the constant terms. So, we have:
(x^3 - 2x^2 + 3x - 1) + (x^2 - x + 4)
Combine like terms:
(x^3 + x^2) + (-2x^2) + (3x - x) + (-1 + 4)
Simplify the exponents and combine like terms again:
x^3 + (-x^2) + 2x - 1 + 4
Simplify further:
x^3 - x^2 + 2x + 3
Therefore, the sum of the two expressions is x^3 - x^2 + 2x + 3.
Step 3: Find the difference (subtraction) of the two expressions.
To subtract the second expression from the first, simply subtract the corresponding coefficients. In other words, subtract the coefficients of the x^3 terms, the x^2 terms, the x terms, and the constant terms. So, we have:
(x^3 - 2x^2 + 3x - 1) - (x^2 - x + 4)
Distribute the negative sign to the terms inside the parenthesis:
x^3 - 2x^2 + 3x - 1 - x^2 + x - 4
Combine like terms:
x^3 + (-2x^2 - x^2) + (3x + x) + (-1 - 4)
Simplify the exponents and combine like terms again:
x^3 - 3x^2 + 4x - 5
Therefore, the difference of the two expressions is x^3 - 3x^2 + 4x - 5.
Step 4: Find the product (multiplication) of the two expressions.
To find the product, we need to multiply each term of the first expression by each term of the second expression. So, we have:
(x^3 - 2x^2 + 3x - 1) * (x^2 - x + 4)
Use the distributive property and multiply each term of the first expression by each term of the second expression:
x^3 * x^2 + x^3 * (-x) + x^3 * 4 + (-2x^2) * x^2 + (-2x^2) * (-x) + (-2x^2) * 4 + 3x * x^2 + 3x * (-x) + 3x * 4 + (-1) * x^2 + (-1) * (-x) + (-1) * 4
Simplify the exponents and combine like terms:
x^5 - x^4 + 4x^3 - 2x^4 + 2x^3 - 8x^2 + 3x^3 - 3x^2 + 12x - x^2 + x - 4
Combine like terms again:
x^5 + (-x^4 - 2x^4) + (4x^3 + 2x^3 + 3x^3) + (-8x^2 - 3x^2 - x^2) + (12x + x) - 4
Simplify further:
x^5 - 3x^4 + 9x^3 - 12x^2 + 13x - 4
Therefore, the product of the two expressions is x^5 - 3x^4 + 9x^3 - 12x^2 + 13x - 4.
To summarize:
- The sum of the two expressions is x^3 - x^2 + 2x + 3.
- The difference of the two expressions is x^3 - 3x^2 + 4x - 5.
- The product of the two expressions is x^5 - 3x^4 + 9x^3 - 12x^2 + 13x - 4.