Question 1
Which expression is equivalent to `x^2 + 3x - 40`?
A) `(x - 4)(x + 10)`
B) `(x - 5)(x + 8)`
C) `(x - 8)(x + 5)`
D) `(x - 10)(x + 4)`
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Question 2
The expression `(24)/(6x + 42)` is equivalent to `4/(x + b)`, where `b` is a constant and `x > 0`. What is the value of `b`?
A) `7`
B) `10`
C) `24`
D) `252`
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Question 3
Which expression is equivalent to `5x⁵ − 6x⁴ + 8x³`?
A) `x⁴(5x − 6)`
B) `x³(5x² − 6x + 8)`
C) `8x³(5x² − 6x + 1)`
D) `6x⁵(−6x⁴ + 8x³ + 1)`
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Question 4
Which expression is equivalent to`(x² + 11)² + (x - 5)(x + 5)`?
A) `x⁴ + 23x² - 14`
B) `x⁴ + 23x² + 96`
C) `x⁴ + 12x² + 121`
D) `x⁴ + x² + 146`
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Question 5
`P = N(19 - C)`
The given equation relates the positive numbers `P`, `N`, and `C`. Which equation correctly expresses `C` in terms of `P` and `N`?
A)`C = (19 + P) / N`
B) `C = (19 - P) / N`
C) `C = 19 + P/N`
D) `C = 19 - P/N`
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Question 1
Which expression is equivalent to `x^2 + 3x - 40`?
A) `(x - 4)(x + 10)`
B) `(x - 5)(x + 8)`
C) `(x - 8)(x + 5)`
D) `(x - 10)(x + 4)`
Answer: B
An expression of the form `x^2 + bx + c`, where `b` and `c` are constants, can be factored if there are two values that add to give `b` and multiply to give `c`. In the given expression, `b = 3` and `c = -40`. The values `-5` and `8` add to give `3` and multiply to give `-40`, so the expression can be factored as `(x - 5)(x + 8)`.
Question 2
The expression `(24)/(6x + 42)` is equivalent to `4/(x + b)`, where `b` is a constant and `x > 0`. What is the value of `b`?
A) `7`
B) `10`
C) `24`
D) `252`
Answer: A
Since the given expressions are equivalent and the numerator of the second expression is `1/6` of the numerator of the first expression, the denominator of the second expression must also be of the denominator of the first expression. By the distributive property, `1/6(6x + 42)` is equivalent to `1/6(6x) + 1/6(42)`, or `x + 7`. Therefore, the value of `b` is `7`.
Question 3
Which expression is equivalent to `5x⁵ − 6x⁴ + 8x³`?
A) `x⁴(5x − 6)`
B) `x³(5x² − 6x + 8)`
C) `8x³(5x² − 6x + 1)`
D) `6x⁵(−6x⁴ + 8x³ + 1)`
Answer: B
To find the equivalent expression, factor out the greatest common factor from each term. The greatest common factor of `5x⁵`, `-6x⁴`, and `8x³` is `x³`. Factoring this out leaves `x³(5x² − 6x + 8)`.
Question 4
Which expression is equivalent to`(x² + 11)² + (x - 5)(x + 5)`?
A) `x⁴ + 23x² - 14`
B) `x⁴ + 23x² + 96`
C) `x⁴ + 12x² + 121`
D) `x⁴ + x² + 146`
Answer: B
First, expand `(x² + 11)²` to get`x⁴ + 22x² + 121`. Next, expand `(x - 5)(x + 5)` (a difference of squares) to get `x² - 25`. Finally, add the two expressions: `(x⁴ + 22x² + 121) + (x² - 25)`. Combine like terms to get `x⁴ + 23x² + 96`.
Question 5
`P = N(19 - C)`
The given equation relates the positive numbers `P`, `N`, and `C`. Which equation correctly expresses `C` in terms of `P` and `N`?
A)`C = (19 + P) / N`
B) `C = (19 - P) / N`
C) `C = 19 + P/N`
D) `C = 19 - P/N`
Answer: D
To isolate `C`, first divide both sides by `N`: `P/N = 19 - C`. Then, subtract `19` from both sides: `(P/N) - 19 = -C`.
Finally, multiply by `-1` to solve for `C`: `C = 19 - P/N`.
Question 1
Which expression is equivalent to `x^2 + 3x - 40`?
A) `(x - 4)(x + 10)`
B) `(x - 5)(x + 8)`
C) `(x - 8)(x + 5)`
D) `(x - 10)(x + 4)`
Question 2
The expression `(24)/(6x + 42)` is equivalent to `4/(x + b)`, where `b` is a constant and `x > 0`. What is the value of `b`?
A) `7`
B) `10`
C) `24`
D) `252`
Question 3
Which expression is equivalent to `5x⁵ − 6x⁴ + 8x³`?
A) `x⁴(5x − 6)`
B) `x³(5x² − 6x + 8)`
C) `8x³(5x² − 6x + 1)`
D) `6x⁵(−6x⁴ + 8x³ + 1)`
Question 4
Which expression is equivalent to`(x² + 11)² + (x - 5)(x + 5)`?
A) `x⁴ + 23x² - 14`
B) `x⁴ + 23x² + 96`
C) `x⁴ + 12x² + 121`
D) `x⁴ + x² + 146`
Question 5
`P = N(19 - C)`
The given equation relates the positive numbers `P`, `N`, and `C`. Which equation correctly expresses `C` in terms of `P` and `N`?
A)`C = (19 + P) / N`
B) `C = (19 - P) / N`
C) `C = 19 + P/N`
D) `C = 19 - P/N`
