Question Video: Factoring a Multivariable Expression by Grouping | Nagwa Question Video: Factoring a Multivariable Expression by Grouping | Nagwa

Question Video: Factoring a Multivariable Expression by Grouping Mathematics • Second Year of Preparatory School

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Factorize fully π₯Β²(π¦ β 5) β 9π₯(π¦ β 5) + 18π¦ β 90.

03:06

Video Transcript

Factorize fully π₯ squared times π¦ minus five minus nine π₯ times π¦ minus five plus 18π¦ minus 90.

It appears that a common factor π₯ squared has been pulled from the first two terms, revealing π¦ minus five in parentheses. And a common factor of negative nine π₯ has been pulled from the second pair of terms, revealing another π¦ minus five in parentheses. In other words, prior to factoring out π₯ squared, the first two terms must have been π₯ squared π¦ minus five π₯ squared. And prior to factoring out negative nine π₯, the second pair of terms must have been negative nine π₯π¦ plus 45π₯.

Here, we can see the given expression written as a polynomial with six terms. The six terms do not have a common factor. However, we have shown that the first pair of terms share a common factor π₯ squared and the second pair of terms share a different common factor negative nine π₯. And we also know that the pairs of factored terms share a common binomial factor of π¦ minus five. This leads us to wonder whether this expression can be factored by grouping.

To determine this, we need to find a common factor of the last two terms. Thankfully, 18π¦ and negative 90 do share a common factor. And thatβs positive 18. When each term has a factor of 18 taken out, we have π¦ minus five in parentheses. And this matches the common factor revealed in the first two pairs of factored terms. This is exactly what we expect to see when factoring by grouping.

In this method, the last step is to factor the expression by identifying a common factor in the factored terms. In this case, that is π¦ minus five. The other factor consists of terms made of the three common factors pulled from each pair of terms, shown here in pink. We are asked to factorize fully, so we must check to see if the quadratic factor is prime or can still be factored.

To factor π₯ squared minus nine π₯ plus 18, we must find two integers that have a product of 18 and a sum of negative nine. Two such numbers exist. They are negative six and negative three. So π₯ squared minus nine π₯ plus 18 is not prime, because the factors are π₯ minus six and π₯ minus three. Altogether, the full factorization of the given expression is π¦ minus five times π₯ minus six times π₯ minus three.

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