Video: Simplifying Rational Expressions

Fully simplify ((π‘₯ + 2)(π‘₯Β² + 7π‘₯ + 12))/((π‘₯ + 7)(π‘₯Β² + 10π‘₯ + 21)).

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Video Transcript

Fully simplify π‘₯ plus two multiplied by π‘₯ squared plus seven π‘₯ plus 12 divided by π‘₯ plus seven multiplied by π‘₯ squared plus 10π‘₯ plus 21.

In order to simplify any algebraic fraction in this form, we firstly need to fully factorize the numerator and denominator and then look to cancel any like terms. Let’s, firstly, consider the quadratic on the numerator, π‘₯ squared plus seven π‘₯ plus 12. In order to factorize any quadratic in this form into two brackets or parentheses, we need to find two numbers that have a product of 12 and a sum of seven. They need to multiply to give us 12 and add to give us seven. There’re three pairs of numbers that have a product of 12, one and 12, two and six, and three and four. Only one of these pairs has a sum of seven, three plus four equals seven. The quadratic π‘₯ squared plus seven π‘₯ plus 12 factorizes into two brackets, π‘₯ plus four multiplied by π‘₯ plus three. These two brackets can be written in either order.

Let’s now consider the quadratic on the denominator, π‘₯ squared plus 10π‘₯ plus 21. This time, we need to find two numbers that have a product of 21 and the sum of 10. There’re two pairs of integers that multiply to give us 21, one and 21, and three and seven. Three plus seven is equal to 10. Therefore, the correct pair is three and seven. The quadratic π‘₯ squared plus 10π‘₯ plus 21 factorizes to π‘₯ plus three multiplied by π‘₯ plus seven.

We can then rewrite the numerator and denominator as a product of three linear factors. The numerator becomes π‘₯ plus two multiplied by π‘₯ plus four multiplied by π‘₯ plus three. The denominator becomes π‘₯ plus seven multiplied by π‘₯ plus three multiplied by π‘₯ plus seven. At this stage, we need to check if any terms on the numerator are the same as the terms on the denominator. π‘₯ plus three is on the top and the bottom. Therefore, we can cancel this out.

The expression simplifies to π‘₯ plus two multiplied by π‘₯ plus four over π‘₯ plus seven multiplied by π‘₯ plus seven. As both terms on the denominator are π‘₯ plus seven, this can be rewritten as π‘₯ plus seven squared. Multiplying any term by itself is the same as squaring this term. The expression π‘₯ plus two multiplied by π‘₯ squared plus seven π‘₯ plus 12 over π‘₯ plus seven multiplied by π‘₯ squared plus 10π‘₯ plus 21, fully simplified, is equal to π‘₯ plus two multiplied by π‘₯ plus four over π‘₯ plus seven squared.

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