### Video Transcript

Scarlett is tiling her bathroom
floor. To find the number of tiles she
needs, she writes the following rational expression, which divides the area of the
room by the area of each tile. 15π₯ squared minus eight π₯ minus
seven divided by π₯ squared minus two π₯ plus one. Write this expression in its
simplest form.

In order to simplify an algebraic
expression in this form, we need to factorize the numerator, factorize the
denominator, and then cancel any common terms. Letβs firstly consider the
quadratic on the denominator, π₯ squared minus two π₯ plus one. As the coefficient of the π₯
squared term is equal to one, this can be factorized into two brackets or
parentheses with first term π₯, as π₯ multiplied by π₯ equals π₯ squared.

The second terms of the two
parentheses need to have a product of positive one and a sum of negative two. The only two pairs of integers that
have a product of positive one are one and one or negative one and negative one. One plus one is equal to two,
whereas negative one plus negative one is equal to negative two. The quadratic π₯ squared minus two
π₯ plus one factorizes to π₯ minus one multiplied by π₯ minus one. This could also be written as π₯
minus one squared.

We now need to consider the
quadratic on the numerator of the fraction, 15π₯ squared minus eight π₯ minus
seven. The first terms in the two
parentheses here need to have a product of 15π₯ squared. This means that we could have five
π₯ and three π₯ or 15π₯ and π₯. The second terms need to have a
product of negative seven. This means that they could be
negative one and seven or negative seven and one.

Once weβve expanded this bracket,
the two π₯ terms need to simplify to negative eight π₯. 15 multiplied by negative one is
equal to negative 15π₯. π₯ multiplied by seven is equal to
seven π₯. Negative 15π₯ plus seven π₯ gives
us negative eight π₯. Our two parentheses are 15π₯ plus
seven and π₯ minus one.

We can check this by expanding the
parentheses using the FOIL method. Multiplying the first terms, 15π₯
and π₯ gives us 15π₯ squared. Multiplying the outside terms gives
us negative 15π₯, as 15π₯ multiplied by negative one is negative 15π₯. Multiplying the inside terms gives
us seven π₯ or positive seven π₯, and multiplying the last terms gives us negative
seven. The two middle terms, the outside
and inside pairs, simplify to negative eight π₯. This proves that the factorization
of 15π₯ squared minus eight π₯ minus seven is 15π₯ plus seven multiplied by π₯ minus
one.

Our initial expression therefore
simplifies to 15π₯ plus seven multiplied by π₯ minus one over π₯ minus one
multiplied by π₯ minus one. As we have an π₯ minus one term on
the numerator and denominator, this can be canceled. The simplest form of the expression
is therefore 15π₯ plus seven divided by π₯ minus one. Scarlett could then use this
simplified version to calculate the number of tiles required for her bathroom
floor.