Video: Simplifying Rational Expressions

Fully simplify (𝑥² + 5𝑥 − 24)/(𝑥² + 15𝑥 + 56).

02:54

Video Transcript

Fully simplify 𝑥 squared plus five 𝑥 minus 24 divided by 𝑥 squared plus 15𝑥 plus 56.

In order to simplify an algebraic expression in this form, we need to factorise the numerator and denominator and then look to cancel any like terms. The quadratic 𝑥 squared plus five 𝑥 minus 24 can be factorized into two brackets or parentheses, where the first term in each bracket is 𝑥. The second terms in each of the parentheses need to have a product of negative 24 and a sum of positive five. There are four pairs of numbers that have a product of 24, one and 24, two and 12, three and eight, and four and six.

In this case, our numbers need to multiply to give negative 24. Therefore, one of the numbers must be positive and one must be negative. Negative three multiplied by positive eight is equal to negative 24. And, negative three plus eight is equal to positive five. The numbers negative three and eight have a product of negative 24 and a sum of positive five. This means that the quadratic 𝑥 squared plus five 𝑥 minus 24 factorises to 𝑥 minus three multiplied by 𝑥 plus eight.

We can repeat this process for the denominator, 𝑥 squared plus 15𝑥 plus 56. We need to find two integers with a product of 56 and a sum of 15. Seven multiplied by eight is equal to 56. And, seven add eight equals 15. Therefore, our two brackets are 𝑥 plus seven and 𝑥 plus eight.

The initial expression simplifies to 𝑥 minus three multiplied by 𝑥 plus eight over 𝑥 plus seven multiplied by 𝑥 plus eight. We have an 𝑥 plus eight term on the numerator and denominator, therefore these will cancel as 𝑥 plus eight divided by 𝑥 plus eight is equal to one. The fully simplified form of 𝑥 squared plus five 𝑥 minus 24 over 𝑥 squared plus 15𝑥 plus 56 is 𝑥 minus three over 𝑥 plus seven.

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