What is the simplest form of the expression (5x + 3xy + 4y) + (4x – 2xy – 2y)?
A. x + xy + 2y
B. x^2 + xy + 2y
C. 9x + 5xy + 2y
D. 9x + 5xy + 6y
E. 9x + xy + 2y
2.
5y + 2z + 3x^2 – (2y – 2z + 4x) is equivalent to:
F. 7y + 4z + 3x2 – 4x
G. 3y + 3x2 – 4x
H. 3y + 4z + 3x2 – 4x
J. 3y + 4z + 7x
K. 3y + 4z + 7x2
3.
Which of the following polynomials is equivalent to (x – 1)(x + 1)(x – 1)?
A. x2 – x + 1
B. x2 – x – 1
C. x3 – x2 – x + 1
D. x3 – x2 – x – 1
E. x3 – 2x2 – 2x + 1
4.
What is the product (2a + 6)2?
F. 2a2 + 12a + 36
G. 4a2 + 12a + 36
H. 4a2 + 36
J. 4a2 + 24a + 36
K. 4a2 – 36
5.
Among the following arithmetic operations, which could the symbol ♣ represent given that the equation (6 ♣ 1)3 – (4 ♣ 1)2 = 200 is true?
I. Subtraction
II. Multiplication
III. Division
A. II only
B. III only
C. II and III only
D. I and III only
E. I, II, and III
no thoughts on any of these?
1. To simplify the given expression (5x + 3xy + 4y) + (4x – 2xy – 2y), we can combine like terms. This means adding or subtracting the coefficients of the same variables.
For the x terms, we have 5x + 4x = 9x.
For the xy terms, we have 3xy - 2xy = xy.
For the y terms, we have 4y - 2y = 2y.
Combining these results, we get 9x + xy + 2y. Therefore, the simplest form of the expression is option E. 9x + xy + 2y.
2. To simplify the expression 5y + 2z + 3x^2 – (2y – 2z + 4x), we need to distribute the negative sign to each term within the parentheses. This means changing the signs of each term.
Expanding the expression, we have 5y + 2z + 3x^2 - 2y + 2z - 4x.
Now, we can combine like terms:
For the y terms, we have 5y - 2y = 3y.
For the z terms, we have 2z + 2z = 4z.
For the x^2 term, we have 3x^2.
For the x term, we have -4x.
Combining these results, we get 3y + 4z + 3x^2 - 4x. Therefore, the simplified expression is option H. 3y + 4z + 3x^2 - 4x.
3. To find the polynomial equivalent to (x – 1)(x + 1)(x – 1), we can simplify each factor and multiply them together.
Expanding each factor:
(x - 1)(x + 1) = x^2 - 1
(x - 1)(x^2 - 1) = x^3 - x^2 - x + 1
Therefore, the polynomial equivalent to (x – 1)(x + 1)(x – 1) is option C. x^3 - x^2 - x + 1.
4. To find the product of (2a + 6)^2, we need to square each term within the parentheses.
Expanding the expression, we have (2a + 6)(2a + 6):
(2a)(2a) + (2a)(6) + (6)(2a) + (6)(6)
= 4a^2 + 12a + 12a + 36
= 4a^2 + 24a + 36
Therefore, the product of (2a + 6)^2 is option J. 4a^2 + 24a + 36.
5. To determine the arithmetic operation represented by the symbol ♣, we can substitute the given equation (6 ♣ 1)3 – (4 ♣ 1)2 = 200 with the available options and see which one makes the equation true.
Let's test each option:
I. Subtraction: (6 - 1)^3 - (4 - 1)^2 = 5^3 - 3^2 = 125 - 9 = 116 (not equal to 200)
II. Multiplication: (6 * 1)^3 - (4 * 1)^2 = 6^3 - 4^2 = 216 - 16 = 200 (matches the equation)
III. Division: (6 ÷ 1)^3 - (4 ÷ 1)^2 = 6^3 - 4^2 = 216 - 16 = 200 (matches the equation)
From the tests, we can see that both multiplication and division operations could be represented by the symbol ♣. Therefore, the answer is option C. II and III only.