Add the polynomial expressions 3−2p−5p2 and p4−3p+4.
A. p4−5p2−5p+7
B. −4p2−5p+7
C. p4−5p2−6p+7
D. −5p2−4p+7
What is (3x2−2)+(11−5x)
A. 2x2−9
B. −2x2+9
C. 3x2−5x+9
D. 3x2+5x−9
The given expression is (3x^2−2)+(11−5x).
To solve this expression, we need to combine like terms.
First, we combine the terms within the parentheses: 3x^2−2+11−5x.
Next, we combine the constant terms: -2+11 = 9.
Finally, we combine the x terms: 3x^2−5x.
Putting it all together, we have 3x^2−5x+9.
Therefore, the answer is C. 3x^2−5x+9.
Which operation shows that polynomials are a closed system under addition?
A. (2x2+6x+1)+(3x2+3x+9)=(2x2+3x+1)+(3x2+6x+9)
B. (2x2+6x+1)+(3x2+3x+9)=(2x2+3x2)+(6x+3x)+(1+9)
C. (2x2+6x+1)+(3x2+3x+9)=5x2+9x+10
D. (2x2+6x+1)+(3x2+3x+9)=−x2+3x−8
To show that polynomials are a closed system under addition, we need to demonstrate that adding two polynomials will always result in another polynomial.
A polynomial is an expression consisting of variables, coefficients, and exponents, with operations of addition and multiplication. When we add two polynomials, we combine like terms, resulting in a new polynomial.
Let's examine the options:
A. (2x^2+6x+1)+(3x^2+3x+9)=(2x^2+3x+1)+(3x^2+6x+9)
This option shows that we can rearrange the terms and add them in any order, which preserves the polynomial structure.
B. (2x^2+6x+1)+(3x^2+3x+9)=(2x^2+3x^2)+(6x+3x)+(1+9)
This option shows how the terms can be combined by adding like terms. The result maintains the polynomial form.
C. (2x^2+6x+1)+(3x^2+3x+9)=5x^2+9x+10
This option shows the correct addition of like terms producing a polynomial with no missing or extraneous terms.
D. (2x^2+6x+1)+(3x^2+3x+9)=−x^2+3x−8
This option shows a different result with a missing term.
Based on the explanations provided, we can conclude that the correct option showing that polynomials are a closed system under addition is:
C. (2x^2+6x+1)+(3x^2+3x+9)=5x^2+9x+10
Which operation shows that polynomials are a closed system under subtraction?
A. (2x2+6x+1)−(3x2+3x+9)=(2x2+6x−4)−(3x2+3x+4)
B. (2x2+6x+1)−(3x2+3x+9)=−x2+3x−8
C. (2x2+6x+1)−(3x2+3x+9)=2x2+6x+1−3x2−3x−9
D. (2x2+6x+1)−(3x2+3x+9)=5x2+9x+10
To show that polynomials are a closed system under subtraction, we need to demonstrate that subtracting two polynomials will always result in another polynomial.
A polynomial is an expression consisting of variables, coefficients, and exponents, with operations of addition, subtraction, and multiplication. When we subtract two polynomials, we combine like terms, resulting in a new polynomial.
Let's examine the options:
A. (2x^2+6x+1)−(3x^2+3x+9)=(2x^2+6x−4)−(3x^2+3x+4)
This option shows how to rearrange the terms and subtract them in any order without affecting the polynomial structure.
B. (2x^2+6x+1)−(3x^2+3x+9)=−x^2+3x−8
This option shows the correct subtraction of like terms producing a polynomial with no missing or extraneous terms.
C. (2x^2+6x+1)−(3x^2+3x+9)=2x^2+6x+1−3x^2−3x−9
This option shows how the terms can be combined by subtracting like terms. The result maintains the polynomial form.
D. (2x^2+6x+1)−(3x^2+3x+9)=5x^2+9x+10
This option shows a different result with missing terms.
Based on the explanations provided, we can conclude that the correct option showing that polynomials are a closed system under subtraction is:
B. (2x^2+6x+1)−(3x^2+3x+9)=−x^2+3x−8
Which operation shows that polynomials are a closed system under addition?
A. (5x2+2x+1)+(−2x2+6x+4)=(5x2−2x2)+(2x+6x)+(1+4)
B. (5x2+2x+1)+(−2x2+6x+4)=(−2x2+2x+4)+(5x2+6x+1)
C. (5x2+2x+1)+(−2x2+6x+4)=−3x2+8x−3
D. (5x2+2x+1)+(−2x2+6x+4)=3x2+8x+5
To add polynomial expressions, we need to combine like terms. Like terms are terms that have the same variable raised to the same exponent.
Let's break down the given polynomial expressions:
3 − 2p − 5p^2
p^4 − 3p + 4
To add these expressions, we need to combine the terms with the same exponent.
Starting with the highest degree term, which is p^4, we have p^4 + 0p^2 + 0p + 0.
Moving on to the next highest degree term, which is -5p^2, we add -5p^2 + 0p + 0.
Then, we combine the terms with the next highest degree, which is -2p, resulting in -2p + 0.
Finally, we add the constant terms, which are 3 + 4, resulting in 7.
Putting it all together, the sum of the polynomial expressions is p^4 - 5p^2 - 2p + 7.
Therefore, the correct answer is C. p^4 - 5p^2 - 2p + 7.
The polynomial expressions are 3−2p−5p^2 and p^4−3p+4.
To add the polynomials, we combine like terms.
The p^4 term is only present in the second polynomial.
The -5p^2 term is present in both polynomials. Its coefficients are -5 and 0, so the sum is -5p^2.
The -2p term is only present in the first polynomial.
The -3p term is present in both polynomials. Its coefficients are 0 and -3, so the sum is -3p.
The 3 and 4 are constants.
The sum of the terms can therefore be written as:
p^4 - 5p^2 - 3p + 3 + 4
Combining the constants, we have:
p^4 - 5p^2 - 3p + 7
Therefore, the sum of the polynomial expressions is:
p^4 - 5p^2 - 3p + 7
Answer: C. p^4 - 5p^2 - 3p + 7
To show that polynomials are a closed system under addition, we need to demonstrate that adding two polynomials will always result in another polynomial.
A polynomial is an expression consisting of variables, coefficients, and exponents, with operations of addition and multiplication. When we add two polynomials, we combine like terms, resulting in a new polynomial.
Let's examine the options:
A. (5x^2+2x+1)+(−2x^2+6x+4)=(5x^2−2x^2)+(2x+6x)+(1+4)
This option shows that we can rearrange the terms and add them in any order, which preserves the polynomial structure.
B. (5x^2+2x+1)+(−2x^2+6x+4)=(−2x^2+2x+4)+(5x^2+6x+1)
This option shows how the terms can be combined by adding like terms. The result maintains the polynomial form.
C. (5x^2+2x+1)+(−2x^2+6x+4)=−3x^2+8x−3
This option shows the correct addition of like terms producing a polynomial with no missing or extraneous terms.
D. (5x^2+2x+1)+(−2x^2+6x+4)=3x^2+8x+5
This option shows a different result with a missing sign.
Based on the explanations provided, we can conclude that the correct option showing that polynomials are a closed system under addition is:
C. (5x^2+2x+1)+(−2x^2+6x+4)=−3x^2+8x−3