Question
Add the polynomial expressions 5z−2z2−4z+5 and 6z2+3−7z .(1 point) Responses 4z2−6z+8 4 z squared minus 6 z plus 8 9z2−11z+8 9 z squared minus 11 z plus 8 4z2−z+8 4 z squared minus z plus 8 11z2+z−2
Answers
Answered by
GPT-4o mini
To add the polynomial expressions \(5z - 2z^2 - 4z + 5\) and \(6z^2 + 3 - 7z\), we can follow these steps:
1. **Combine like terms from the first polynomial**:
- Combine \(5z - 4z\) which gives \(1z\) or just \(z\).
- The first polynomial simplifies to:
\[
-2z^2 + z + 5
\]
2. **Now, write the second polynomial**:
\[
6z^2 - 7z + 3
\]
3. **Add the two simplified polynomials**:
Combine the coefficients of like terms:
- For \(z^2\): \(-2z^2 + 6z^2 = 4z^2\)
- For \(z\): \(z - 7z = -6z\)
- For the constant term: \(5 + 3 = 8\)
So, putting it all together, we get:
\[
4z^2 - 6z + 8
\]
Therefore, the answer is: **4z² - 6z + 8**.
1. **Combine like terms from the first polynomial**:
- Combine \(5z - 4z\) which gives \(1z\) or just \(z\).
- The first polynomial simplifies to:
\[
-2z^2 + z + 5
\]
2. **Now, write the second polynomial**:
\[
6z^2 - 7z + 3
\]
3. **Add the two simplified polynomials**:
Combine the coefficients of like terms:
- For \(z^2\): \(-2z^2 + 6z^2 = 4z^2\)
- For \(z\): \(z - 7z = -6z\)
- For the constant term: \(5 + 3 = 8\)
So, putting it all together, we get:
\[
4z^2 - 6z + 8
\]
Therefore, the answer is: **4z² - 6z + 8**.
Answer
Which of the following correctly simplifies (9x2+17x+13)−(2x2+12x+11) ?(1 point) Responses 7x2+5x+2 7 x squared plus 5 x plus 2 7x2+29x+24 7 x squared plus 29 x plus 24 9x2+17x+13−2x2−12x−11 9 x squared plus 17 x plus 13 minus 2 x squared minus 12 x minus 11 12x2+2
Answered by
GPT-4o mini
To simplify the expression \((9x^2 + 17x + 13) - (2x^2 + 12x + 11)\), we proceed as follows:
1. **Distribute the negative sign** into the second expression:
\[
(9x^2 + 17x + 13) - (2x^2 + 12x + 11) = 9x^2 + 17x + 13 - 2x^2 - 12x - 11
\]
2. **Combine like terms**:
- For \(x^2\) terms:
\[
9x^2 - 2x^2 = 7x^2
\]
- For \(x\) terms:
\[
17x - 12x = 5x
\]
- For constant terms:
\[
13 - 11 = 2
\]
3. **Combine all results**:
\[
7x^2 + 5x + 2
\]
Therefore, the correct simplification is: **7x² + 5x + 2**.
1. **Distribute the negative sign** into the second expression:
\[
(9x^2 + 17x + 13) - (2x^2 + 12x + 11) = 9x^2 + 17x + 13 - 2x^2 - 12x - 11
\]
2. **Combine like terms**:
- For \(x^2\) terms:
\[
9x^2 - 2x^2 = 7x^2
\]
- For \(x\) terms:
\[
17x - 12x = 5x
\]
- For constant terms:
\[
13 - 11 = 2
\]
3. **Combine all results**:
\[
7x^2 + 5x + 2
\]
Therefore, the correct simplification is: **7x² + 5x + 2**.
Answer
Multiply the polynomials (3x−1)(2x+4) .
Answered by
GPT-4o mini
To multiply the polynomials \((3x - 1)(2x + 4)\), we can use the distributive property (also known as the FOIL method for binomials):
1. **Multiply the first terms**:
\[
3x \cdot 2x = 6x^2
\]
2. **Multiply the outer terms**:
\[
3x \cdot 4 = 12x
\]
3. **Multiply the inner terms**:
\[
-1 \cdot 2x = -2x
\]
4. **Multiply the last terms**:
\[
-1 \cdot 4 = -4
\]
5. **Combine all the products**:
\[
6x^2 + 12x - 2x - 4
\]
6. **Combine like terms**:
\[
6x^2 + (12x - 2x) - 4 = 6x^2 + 10x - 4
\]
The final result of the multiplication is:
\[
\boxed{6x^2 + 10x - 4}
\]
1. **Multiply the first terms**:
\[
3x \cdot 2x = 6x^2
\]
2. **Multiply the outer terms**:
\[
3x \cdot 4 = 12x
\]
3. **Multiply the inner terms**:
\[
-1 \cdot 2x = -2x
\]
4. **Multiply the last terms**:
\[
-1 \cdot 4 = -4
\]
5. **Combine all the products**:
\[
6x^2 + 12x - 2x - 4
\]
6. **Combine like terms**:
\[
6x^2 + (12x - 2x) - 4 = 6x^2 + 10x - 4
\]
The final result of the multiplication is:
\[
\boxed{6x^2 + 10x - 4}
\]
Answer
Which of the following responses shows that polynomials form a closed system under multiplication?(1 point) Responses 3x24−1 Start Fraction 3 x squared over 4 minus 1 End Fraction 14x3⋅5 Start Fraction 1 over 4 x cubed End Fraction times 5 (x)(12x) left parenthesis x right parenthesis left parenthesis Start Fraction 1 over 2 End Fraction x right parenthesis (x−−√)(6x ⋅−3 )
Answered by
GPT-4o mini
To demonstrate that polynomials form a closed system under multiplication, we need to show that when we multiply two polynomials, the result is also a polynomial.
Let's analyze the provided responses:
1. **\(3x^2 - 1\)**: This is a polynomial. If you multiply it by another polynomial, you get another polynomial.
2. **\(\frac{3x^2}{4} - 1\)**: This is a polynomial (a constant multiple of a polynomial). Multiplying it by another polynomial yields a polynomial.
3. **\(14x^3 \cdot 5\)**: This simplifies to \(70x^3\), which is also a polynomial.
4. **\((x)(12x)\)**: This product simplifies to \(12x^2\), which is a polynomial.
5. **\(\left(x\right)\left(\frac{1}{2}x\right)\)**: This results in \(\frac{1}{2}x^2\), which is also a polynomial.
6. **\((x^{1/2})(6x \cdot -3)\)**: \(x^{1/2}\) is not a polynomial because polynomials can only have non-negative integer powers.
From these responses, the examples that clearly demonstrate polynomial multiplication yielding another polynomial are:
- \(3x^2 - 1\)
- \(14x^3 \cdot 5\)
- \((x)(12x)\)
- \(\left(x\right)\left(\frac{1}{2}x\right)\)
However, the option that best illustrates the concept that multiplication of polynomials yields another polynomial is the response \( (x)(12x) \), as it directly represents the multiplication of two polynomial terms resulting in another polynomial without any additional information.
Thus, the answer that best illustrates that polynomials form a closed system under multiplication is **\((x)(12x)\)**.
Let's analyze the provided responses:
1. **\(3x^2 - 1\)**: This is a polynomial. If you multiply it by another polynomial, you get another polynomial.
2. **\(\frac{3x^2}{4} - 1\)**: This is a polynomial (a constant multiple of a polynomial). Multiplying it by another polynomial yields a polynomial.
3. **\(14x^3 \cdot 5\)**: This simplifies to \(70x^3\), which is also a polynomial.
4. **\((x)(12x)\)**: This product simplifies to \(12x^2\), which is a polynomial.
5. **\(\left(x\right)\left(\frac{1}{2}x\right)\)**: This results in \(\frac{1}{2}x^2\), which is also a polynomial.
6. **\((x^{1/2})(6x \cdot -3)\)**: \(x^{1/2}\) is not a polynomial because polynomials can only have non-negative integer powers.
From these responses, the examples that clearly demonstrate polynomial multiplication yielding another polynomial are:
- \(3x^2 - 1\)
- \(14x^3 \cdot 5\)
- \((x)(12x)\)
- \(\left(x\right)\left(\frac{1}{2}x\right)\)
However, the option that best illustrates the concept that multiplication of polynomials yields another polynomial is the response \( (x)(12x) \), as it directly represents the multiplication of two polynomial terms resulting in another polynomial without any additional information.
Thus, the answer that best illustrates that polynomials form a closed system under multiplication is **\((x)(12x)\)**.
Answer
Structures of Expressions Unit Test 13 of 1713 of 17 Items Question Using the Commutative Property, prove that these algebraic expressions are equivalent. In 1–2 sentences, explain your result. Expression 1: y⋅3x Expression 2: 3xy
Answered by
GPT-4o mini
To prove that the algebraic expressions \(y \cdot 3x\) and \(3xy\) are equivalent using the Commutative Property, we can rearrange the terms in the first expression. According to the Commutative Property of multiplication, the order of factors does not change the product. Therefore,
\[
y \cdot 3x = 3 \cdot y \cdot x = 3xy
\]
This shows that both expressions represent the same mathematical value, thus proving they are equivalent.
**Explanation**: The Commutative Property allows us to change the order of multiplication, confirming that \(y \cdot 3x\) is equivalent to \(3xy\).
\[
y \cdot 3x = 3 \cdot y \cdot x = 3xy
\]
This shows that both expressions represent the same mathematical value, thus proving they are equivalent.
**Explanation**: The Commutative Property allows us to change the order of multiplication, confirming that \(y \cdot 3x\) is equivalent to \(3xy\).
Answer
Use the Associative Property to demonstrate that Expression 1 is equivalent to Expression 2. Expression 1: 22r−13 Expression 2: 19r+(3r−2)−11
Answered by
GPT-4o mini
To use the Associative Property to demonstrate that the expression \(22r - 13\) (Expression 1) is equivalent to the expression \(19r + (3r - 2) - 11\) (Expression 2), we need to simplify Expression 2.
Starting with Expression 2:
\[
19r + (3r - 2) - 11
\]
1. **First, simplify the expression inside the parentheses**:
\[
3r - 2
\]
2. **Now, rewrite Expression 2 without parentheses**:
\[
19r + 3r - 2 - 11
\]
3. **Use the Associative Property to group like terms**:
\[
(19r + 3r) + (-2 - 11)
\]
4. **Combine the like terms**:
- For the \(r\) terms:
\[
19r + 3r = 22r
\]
- For the constants:
\[
-2 - 11 = -13
\]
So, we have:
\[
22r - 13
\]
Now we see that Expression 1 is indeed equal to Expression 2:
\[
22r - 13 = 22r - 13
\]
### Explanation:
By using the Associative Property, we were able to group and rearrange terms in Expression 2 to simplify it to \(22r - 13\), demonstrating that both expressions are equivalent.
Starting with Expression 2:
\[
19r + (3r - 2) - 11
\]
1. **First, simplify the expression inside the parentheses**:
\[
3r - 2
\]
2. **Now, rewrite Expression 2 without parentheses**:
\[
19r + 3r - 2 - 11
\]
3. **Use the Associative Property to group like terms**:
\[
(19r + 3r) + (-2 - 11)
\]
4. **Combine the like terms**:
- For the \(r\) terms:
\[
19r + 3r = 22r
\]
- For the constants:
\[
-2 - 11 = -13
\]
So, we have:
\[
22r - 13
\]
Now we see that Expression 1 is indeed equal to Expression 2:
\[
22r - 13 = 22r - 13
\]
### Explanation:
By using the Associative Property, we were able to group and rearrange terms in Expression 2 to simplify it to \(22r - 13\), demonstrating that both expressions are equivalent.