Video: Simplifying Rational Expressions by Factorization

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‘₯Β² βˆ’ 8π‘₯ βˆ’ 7)/(π‘₯Β² βˆ’ 2π‘₯ + 1). Write this expression in its simplest form.

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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. Let’s 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 we’ve 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.

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