Simplify the rational expressions below: x 2 + 5 x + 6 x 2 + 7 x + 10 \frac{x^2+5x+6}{x^2+7x+10}

this problem, we're asked to simplify the rational expression below. Now notice that we have these quadratics in the numerator and denominator of our rational expression. A good first step to solve any of these simplifying problems is to factor the numerator and denominator. Now what I'm gonna start by doing here is factoring the numerator. Now I can factor this since I can see that there is a leading coefficient of one in front of the x squared. All I need to do is think about a pair of numbers that multiply to give me six and add to give me five. Now thinking about that pair of numbers, that's going to be two and three. Two times three is six, and two plus three is five. So this is going to factor into x plus two times x plus three. So that's what we get in our numerator. Now I also need to factor the denominator. So what I need to do is think about a pair of numbers that multiply to give me 10, and add to give me seven. And this will work since again I can see there's a one coefficient that's leading in front of the x squared. Well thinking about that, that's going to be five and two, because five times two that gives me ten, and five plus two that gives me seven. So this factors into x plus five times x plus two. So notice we have factored the numerator and we factored the denominator, and I can see there is an x plus two that I can cancel in both. So simplifying this rational expression, we get x plus three on top, and then we get x plus five on the bottom. So x plus three over x plus five would be the solution to this problem.

8. Rational Expressions and Equations

Simplifying Rational Expressions

Beginning Algebra

8. Rational Expressions and Equations

Simplifying Rational Expressions

Struggling with Beginning Algebra?

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Multiple Choice

Simplify the rational expressions below:
$\frac{x^{2} + 5 x + 6}{x^{2} + 7 x + 10}$

$\frac{x + 3}{x + 2}$

$\frac{x + 3}{x + 5}$

$\frac{x + 2}{x + 2}$

$1$

Verified step by step guidance

Start by factoring both the numerator and the denominator of the rational expression $\frac{x^2 + 5x + 6}{x^2 + 7x + 10}$ into products of binomials. Look for two numbers that multiply to the constant term and add to the coefficient of the middle term in each quadratic.

For the numerator $x^2 + 5x + 6$, find two numbers that multiply to 6 and add to 5. Write the numerator as a product of two binomials using these numbers.

For the denominator $x^2 + 7x + 10$, find two numbers that multiply to 10 and add to 7. Write the denominator as a product of two binomials using these numbers.

After factoring, write the rational expression as $\frac{(x + a)(x + b)}{(x + c)(x + d)}$ where $a$, $b$, $c$, and $d$ are the numbers found in the previous steps.

Look for any common binomial factors in the numerator and denominator and cancel them out to simplify the expression to its lowest terms.

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Struggling with Beginning Algebra?

Simplify the rational expressions below:x2+5x+6x2+7x+10\frac{x^2+5x+6}{x^2+7x+10}

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Simplify the rational expressions below:
$\frac{x^{2} + 5 x + 6}{x^{2} + 7 x + 10}$