Final answer:
The sum of the polynomials (9 – 3x^2) and (–8x^2 + 4x + 5) is found by combining like terms to get the result 14 – 11x^2 + 4x. The properties of arithmetic, such as the commutative property of addition, apply while adding polynomial terms. In multiplying terms with the same base, exponents are added.
Step-by-step explanation:
To find the sum of the polynomials (9 – 3x^2) + (–8x^2 + 4x + 5), we should simply add together the like terms. The like terms are the terms that have the same variables raised to the same power. In the given expressions, the like terms are the ones with x^2 and the constant terms (the numbers without variables).
Here's a step-by-step breakdown:
- Combine the constant terms: 9 + 5 = 14.
- Combine the x^2 terms: –3x^2 + (–8x^2) = –11x^2.
- The x term in the second polynomial does not have a like term in the first polynomial, so we simply bring it down: 4x.
Putting it all together, the sum of the polynomials is 14 – 11x^2 + 4x.
The commutative property of addition, which states that the order in which you add numbers does not affect the sum, applies here. For multiplication, similar rules apply: when multiplying terms with the same base, we add the exponents, as seen in the mathematical rule x^p × x^q = x^(p+q).
It's also important to remember basic arithmetic properties when combining terms:
- Adding two positive numbers yields a positive result.
- Adding two negative numbers yields a negative result.
- When numbers with opposite signs are added, subtract the smaller number from the larger number, giving the result the sign of the larger number.
- Multiplication and division follow similar rules regarding the signs of the numbers involved.