Question

choose the correct answer

1. if (3x^2 - 5x + 2 = 0) and (x = 1), what is the value of the expression?

a. 0
b. 1
c. -1
d. 2

1. evaluate (2a^3 - 3a^2 + 4a - 5) when (a = -1).

a. -14
b. 14
c. -4
d. 4

1. if (p(x) = x^3 + 2x^2 - 5x + 3), find (p(2) - p(1)).

a. 6
b. 8
c. 10
d. 12

1. simplify:

\\\frac{3x^2+6x}{x+2} - \frac{x^2-4}{x+2}\\
a. (2x - 2)
b. (2x + 2)

Explanation:

🆕 New Concept Discovered: Algebraic Evaluation and Simplification
Evaluating expressions by substitution and simplifying rational expressions.

Step 1: Solve Question 1

We need to find the value of the expression \( 3x^2 - 5x + 2 \) when \( x = 1 \).

Substitute \( x = 1 \) into the expression:

$$ 3(1)^2 - 5(1) + 2 $$

$$ = 3(1) - 5 + 2 $$

$$ = 3 - 5 + 2 $$

$$ = 0 $$

Thus, the correct option is A.

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Step 2: Solve Question 2

We need to evaluate \( 2a^3 - 3a^2 + 4a - 5 \) when \( a = -1 \).

Substitute \( a = -1 \) into the expression:

$$ 2(-1)^3 - 3(-1)^2 + 4(-1) - 5 $$

Evaluate each term:

• \( (-1)^3 = -1 \), so \( 2(-1)^3 = -2 \)
• \( (-1)^2 = 1 \), so \( -3(-1)^2 = -3 \)
• \( 4(-1) = -4 \)

Combine the terms:

$$ -2 - 3 - 4 - 5 = -14 $$

Thus, the correct option is A.

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Step 3: Solve Question 3

We are given \( P(x) = x^3 + 2x^2 - 5x + 3 \). We need to find \( P(2) - P(1) \).

First, calculate \( P(2) \):

$$ P(2) = (2)^3 + 2(2)^2 - 5(2) + 3 $$

$$ P(2) = 8 + 2(4) - 10 + 3 $$

$$ P(2) = 8 + 8 - 10 + 3 = 9 $$

Next, calculate \( P(1) \):

$$ P(1) = (1)^3 + 2(1)^2 - 5(1) + 3 $$

$$ P(1) = 1 + 2 - 5 + 3 = 1 $$

Now, find the difference:

$$ P(2) - P(1) = 9 - 1 = 8 $$

Thus, the correct option is B.

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Step 4: Solve Question 4

We need to simplify the expression:

$$ \frac{3x^2 + 6x}{x + 2} - \frac{x^2 - 4}{x + 2} $$

Since the denominators are the same, combine the numerators over the common denominator:

$$ \frac{(3x^2 + 6x) - (x^2 - 4)}{x + 2} $$

$$ = \frac{3x^2 + 6x - x^2 + 4}{x + 2} $$

$$ = \frac{2x^2 + 6x + 4}{x + 2} $$

Factor the numerator:

$$ 2x^2 + 6x + 4 = 2(x^2 + 3x + 2) $$

$$ = 2(x + 1)(x + 2) $$

Substitute this back into the expression:

$$ \frac{2(x + 1)(x + 2)}{x + 2} $$

Cancel the common factor \( (x + 2) \) (assuming \( x
eq -2 \)):

$$ 2(x + 1) = 2x + 2 $$

Thus, the correct option is B.

Answer:

1. A. 0
2. A. -14
3. B. 8
4. B. 2x + 2