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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.
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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.
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Letβs, firstly, consider the quadratic on the numerator, π₯ squared plus seven π₯ plus 12.
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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.
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They need to multiply to give us 12 and add to give us seven.
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Thereβre three pairs of numbers that have a product of 12, one and 12, two and six, and three and four.
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Only one of these pairs has a sum of seven, three plus four equals seven.
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The quadratic π₯ squared plus seven π₯ plus 12 factorizes into two brackets, π₯ plus four multiplied by π₯ plus three.
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These two brackets can be written in either order.
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Letβs now consider the quadratic on the denominator, π₯ squared plus 10π₯ plus 21.
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This time, we need to find two numbers that have a product of 21 and the sum of 10.
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Thereβre two pairs of integers that multiply to give us 21, one and 21, and three and seven.
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Three plus seven is equal to 10.
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Therefore, the correct pair is three and seven.
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The quadratic π₯ squared plus 10π₯ plus 21 factorizes to π₯ plus three multiplied by π₯ plus seven.
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We can then rewrite the numerator and denominator as a product of three linear factors.
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The numerator becomes π₯ plus two multiplied by π₯ plus four multiplied by π₯ plus three.
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The denominator becomes π₯ plus seven multiplied by π₯ plus three multiplied by π₯ plus seven.
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At this stage, we need to check if any terms on the numerator are the same as the terms on the denominator.
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π₯ plus three is on the top and the bottom.
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Therefore, we can cancel this out.
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The expression simplifies to π₯ plus two multiplied by π₯ plus four over π₯ plus seven multiplied by π₯ plus seven.
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As both terms on the denominator are π₯ plus seven, this can be rewritten as π₯ plus seven squared.
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Multiplying any term by itself is the same as squaring this term.
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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.