Simplify each expression. x 2 − 10 x + 24 ( 4 − x ) ( 6 − x ) \frac{x^2-10x+24}{(4-x)(6-x)}

this problem, we need to simplify the expression. And let's take a look at the rational expression that we're given. X squared minus 10 x plus 24 over four minus x times six minus x. Now right off the bat, there's not really a simple way that we can think about simplifying what we have. So what I'm instead going to do is I'm going to factor the top of this expression. The denominator is going to stay the same, so we're just gonna keep that four minus x, and that six minus x. But let's see what we can do to the top. Well, what I have on top is x squared minus 10 x plus 24. This is a quadratic, and we should know how to factor quadratics. When we have one as the leading coefficient, we need to find a pair of numbers that multiplies to give us the 24 and adds to give us negative 10. So what exactly is this going to be? Well, Well I could go through every possible pair of numbers that multiplies to give me 24, positive and negative, and it turns out what's going to work is negative four and negative six. Because negative four times negative six, the negatives cancel giving us positive 24, And then adding these will give us of course negative 10. So what we're going to end up with is breaking this down into x minus four times x minus six. So this is what we get. Now I want you to notice something interesting that happens. We have opposite factors x minus four and four minus x. We also have opposite factors x minus six and six minus x. What is this going to look like? Well, think of these as two separate fractions being multiplied together. Think of this as x minus four over four minus x as a fraction being multiplied by x minus six over six minus x as a fraction there. Think about what this is gonna look like, because we already know when dealing with the opposite factors, that gives us negative one. We know when dealing with these opposite factors, that gives us negative one as well. So we end up with negative one times negative one, which is just going to give us positive one. So all of this expression that we see simply breaks down into positive one as the solution.

Simplify each expression. x 2 − 10 x + 24 ( 4 − x ) ( 6 − x ) \frac{x^2-10x+24}{(4-x)(6-x)}

( 4 − x ) ( 6 − x ) (4-x)(6-x) ( 4 − x ) ( 6 − x )

8. Rational Expressions and Equations

Simplifying Rational Expressions

Beginning & Intermediate Algebra

8. Rational Expressions and Equations

Simplifying Rational Expressions

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Multiple Choice

Simplify each expression.
$\frac{x^{2} - 10 x + 24}{(4 - x) (6 - x)}$

$1$

$-1$

$0$

$(4-x)(6-x)$

Verified step by step guidance

Start by factoring the quadratic expression in the numerator: $x^2 - 10x + 24$. Look for two numbers that multiply to 24 and add up to -10.

Rewrite the numerator as a product of two binomials using the numbers found in the previous step, so it looks like $(x - a)(x - b)$ where $a$ and $b$ are the numbers from step 1.

Rewrite the denominator $(4 - x)(6 - x)$ by factoring out a negative sign from each binomial to express them in the form $(x - 4)$ and $(x - 6)$, if possible, to match the factors in the numerator.

After rewriting, identify and cancel out any common factors that appear in both the numerator and denominator.

Write the simplified expression after canceling common factors, which should be the simplest form of the original fraction.

4:37m

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Simplify each expression.x2−10x+24(4−x)(6−x)\(\frac{x^2-10x+24}{(4-x)(6-x)}\)

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Simplify each expression.
$\frac{x^{2} - 10 x + 24}{(4 - x) (6 - x)}$