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Fully simplify 𝑥 squared plus five 𝑥 minus 24 divided by 𝑥 squared plus 15𝑥 plus 56.
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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.
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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.
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The second terms in each of the parentheses need to have a product of negative 24 and a sum of positive five.
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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.
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In this case, our numbers need to multiply to give negative 24.
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Therefore, one of the numbers must be positive and one must be negative.
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Negative three multiplied by positive eight is equal to negative 24.
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And, negative three plus eight is equal to positive five.
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The numbers negative three and eight have a product of negative 24 and a sum of positive five.
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This means that the quadratic 𝑥 squared plus five 𝑥 minus 24 factorises to 𝑥 minus three multiplied by 𝑥 plus eight.
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We can repeat this process for the denominator, 𝑥 squared plus 15𝑥 plus 56.
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We need to find two integers with a product of 56 and a sum of 15.
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Seven multiplied by eight is equal to 56.
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And, seven add eight equals 15.
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Therefore, our two brackets are 𝑥 plus seven and 𝑥 plus eight.
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The initial expression simplifies to 𝑥 minus three multiplied by 𝑥 plus eight over 𝑥 plus seven multiplied by 𝑥 plus eight.
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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.
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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.