(a^(2)-b^(2)+2a+1)/(a+1+b)= ? A) a+3b+1 B) 3a-b+1 C) a-b+1 D) a-b+3 E) a+b-1

Question

(a^(2)-b^(2)+2a+1)/(a+1+b)= ? A)  a+3b+1 B)  3a-b+1 C)  a-b+1 D)  a-b+3 E)  a+b-1

Bytelearn · Accepted Answer

Recognize Quadratic Expression: First, recognize that the numerator is a quadratic expression that can be factored. The term a^2 - b^2 is a difference of squares, which factors into (a - b)(a + b). The remaining terms 2a + 1 do not factor with the difference of squares, so we will keep them separate for now.Factor Numerator: Now, let's factor the numerator: a^2 - b^2 + 2a + 1 = (a - b)(a + b) + 2a + 1.Look for Common Factors: Next, we need to look for common factors between the numerator and the denominator. The denominator is a + 1 + b, which can be reordered as (a + b + 1) to make it easier to compare with the numerator.Expand and Simplify Numerator: We can see that there is no common factor between (a - b)(a + b) + 2a + 1 and a + b + 1, so we cannot simplify the expression by canceling out any terms. Therefore, we need to expand the numerator and then simplify the expression.Factor Perfect Square Trinomial: Let's expand the numerator: (a - b)(a + b) + 2a + 1 = a^2 - b^2 + 2a + 1.Rewrite with Factored Numerator: Now, we will simplify the numerator by combining like terms: a^2 - b^2 + 2a + 1 = a^2 + 2a + 1 - b^2.Simplify by Canceling Common Factor: Notice that a^2 + 2a + 1 is a perfect square trinomial, which factors into (a + 1)^2. So the numerator becomes (a + 1)^2 - b^2.Final Simplified Expression: We can now rewrite the expression with the factored numerator: ((a + 1)^2 - b^2) / (a + b + 1).The numerator is now a difference of squares again, with (a + 1)^2 being the square of (a + 1) and b^2 being the square of b. We can factor the numerator as ((a + 1) - b)((a + 1) + b).After factoring, the expression becomes: ((a + 1 - b)(a + 1 + b)) / (a + b + 1).Now, we can simplify the expression by canceling out the common factor of (a + b + 1) in the numerator and denominator. However, we must be careful here because the numerator has (a + 1 - b) and (a + 1 + b), which are not the same as (a + b + 1). There is no common factor to cancel out, so the expression cannot be simplified further.The final simplified expression is ((a + 1 - b)(a + 1 + b)) / (a + b + 1), which does not match any of the answer choices provided. It seems there might have been a mistake in the simplification process. Let's re-evaluate the expression.

$40$ . $\frac{a^{2}-b^{2}+2 a+1}{a+1+b}=$ ? $\newline$ A) $a+3 b+1$ $\newline$ B) $3 a-b+1$ $\newline$ C) $a-b+1$ $\newline$ D) $a-b+3$ $\newline$ E) $a+b-1$

Full solution

More problems from Factor sums and differences of cubes

Related topics

404040. a2−b2+2a+1a+1+b= \frac{a^{2}-b^{2}+2 a+1}{a+1+b}= a+1+ba2−b2+2a+1​= ?\newlineA) a+3b+1 a+3 b+1 a+3b+1\newlineB) 3a−b+1 3 a-b+1 3a−b+1\newlineC) a−b+1 a-b+1 a−b+1\newlineD) a−b+3 a-b+3 a−b+3\newlineE) a+b−1 a+b-1 a+b−1

$40$ . $\frac{a^{2}-b^{2}+2 a+1}{a+1+b}=$ ? $\newline$ A) $a+3 b+1$ $\newline$ B) $3 a-b+1$ $\newline$ C) $a-b+1$ $\newline$ D) $a-b+3$ $\newline$ E) $a+b-1$