Question

a gym floor has an area of $x^2 + 5x - 36$ square feet. is it possible for one of the dimensions to be $(x - 9)$ feet? explain.

Explanation:

Step1: Factor the quadratic expression

To determine if \((x - 9)\) can be a factor of \(x^{2}+5x - 36\), we try to factor the quadratic. We need two numbers that multiply to \(- 36\) and add up to \(5\). Let's find those numbers. The factors of \(-36\) are: \((-1,36),(1, - 36),(-2,18),(2,-18),(-3,12),(3,-12),(-4,9),(4,-9),(-6,6)\). The pair that adds up to \(5\) is \(9\) and \(-4\) (since \(9+( - 4)=5\) and \(9\times(-4)=-36\)). So we can factor \(x^{2}+5x - 36\) as:
\[x^{2}+5x - 36=(x + 9)(x-4)\]

Step2: Compare with the given factor

We were asked if \((x - 9)\) can be a factor. From the factored form \((x + 9)(x - 4)\), the factors are \((x + 9)\) and \((x - 4)\), not \((x - 9)\). Also, if we use polynomial long - division or the factor theorem: The factor theorem states that if \((x - a)\) is a factor of a polynomial \(f(x)\), then \(f(a)=0\). Let \(f(x)=x^{2}+5x - 36\). If \((x - 9)\) is a factor, then \(f(9)=0\). Calculate \(f(9)\):
\[f(9)=9^{2}+5\times9 - 36=81 + 45-36=90
eq0\]
So \((x - 9)\) is not a factor of \(x^{2}+5x - 36\), which means it is not possible for one of the dimensions to be \((x - 9)\) feet.

Answer:

It is not possible for one of the dimensions to be \((x - 9)\) feet. When we factor the quadratic expression \(x^{2}+5x - 36\), we get \((x + 9)(x - 4)\) (by finding two numbers that multiply to \(-36\) and add to \(5\), which are \(9\) and \(-4\)). Also, using the factor theorem, if we substitute \(x = 9\) into \(x^{2}+5x - 36\), we get \(9^{2}+5\times9-36=81 + 45 - 36=90
eq0\), so \((x - 9)\) is not a factor of the quadratic representing the area of the gym floor.