Підходи до спрощення раціональних виразів

(Videos) Reducing Rational Expressions

by mathman1024

Here's a video by mathman1024 showing you how to reduce rational expressions.

$\large{\text{Summary}}$

A rational expression is the quotient of two polynomials, $\frac{P}Q$, where $Q\ne0$.

To reduce or simply you just look for common factors and cancel. For example he looks at the fraction, $\frac{12}{33}$. This is a rational expression and we reduce it by breaking it down into factors. $$\begin{align} \frac{12}{33}&=\frac{\color{#8925AE}{3}\color{black}{\cdot4}}{\color{#8925AE}{3}\color{black}{\cdot11}}\\ &=\color{#8925AE}{\frac{3}{3}}\color{black}{\cdot\frac4{11}}\\ &=\color{#8925AE}{1}\color{black}{\cdot\frac4{11}}\\ &=\frac4{11} \end{align}$$

Now let's look at more complicated example, $\frac{x^2+7x+10}{x^2-x-30}$. The first step is to factor both quadratics using whatever technique you like. $$\begin{align} \frac{x^2+7x+10}{x^2-x-30}&=\frac{\color{#8925AE}{(x+5)}\color{black}{(x+2)}}{\color{#8925AE}{(x+5)}\color{black}{(x-6)}}\\ &=\frac{\color{#8925AE}{\cancel{(x+5)}}\color{black}{(x+2)}}{\color{#8925AE}{\cancel{(x+5)}}\color{black}{(x-6)}}\\ &=\frac{x+2}{x-6} \end{align}$$

Here's another video by mathman1024 with some more rational expressions to simplify.

The first one he looks at is, $\frac{24x^3-18x^2}{15x^4+21x^3}$. The first step is to remove the greatest common factor from both the numerator and denominator. $$\begin{align} \frac{24x^3-18x^2}{15x^4+21x^3}&=\frac{6x^2(4x-3)}{3x^3(5x+7)} \end{align}$$ We can now simplify the fraction we just removed, $\frac{6x^2}{3x^3}$. $$\begin{align} \dfrac{6x^2}{3x^3}&=2x^{2-3}\\ &=2x^{-1}\\ &=\frac2x \end{align}$$

So our rational expression becomes, $$\begin{align} \frac{24x^3-18x^2}{15x^4+21x^3}&=\frac{6x^2(4x-3)}{3x^3(5x+7)}\\ &=\frac{2(4x-3)}{x(5x+7)} \end{align}$$ This is the furthest you can reduce. Our denominator is still a polynomial.

Try this next one on your own!

$$\begin{align} \frac{2x^2-5x+2}{4x^2-1}&=\frac{(2x-1)(x-2)}{(2x+1)(2x-1)}\\ &=\frac{\cancel{(2x-1)}(x-2)}{\cancel{(2x-1)}(2x+1)}\\ &=\boxed{\frac{x-2}{2x+1}} \end{align}$$

Rational expressions combine skills from fractions and solving equations

A fraction is a number with a top part and a bottom part. The top number is called the numerator, and the bottom number is called the denominator.

A rational number is any number that can be written as a fraction where both the numerator and denominator are integers.

A rational expression is similar to a fraction, but it includes at least one variable in the numerator or denominator. Sometimes there may also be addition or subtraction in the top or bottom.

Working with rational expressions is a lot like simplifying regular fractions. You follow similar steps, but there is a little more algebra involved.

Rational equations are just slightly different - rational equations have an equal sign, but expressions do not.

Fractions: $$\frac{2}{-3}, \frac{-x}{5}, \frac{5}{x-2}, \frac{3y}{100}, \frac{2+y}{x}$$

Rational numbers: $$\frac{2}{1}, \frac{8}{9}, \frac{55}{-2}, \frac{22}{5}, \frac{212}{1000}$$

Rational expressions: $$\frac{2x+3}{x^2}, \frac{x}{9}, \frac{55x^2+2xy+17}{-2x}, \frac{22-x+y}{5-xy}, \frac{200}{200+8x}$$

You can do arithmetic on rational expressions the same way you do arithmetic on fractions. Let's work through an example together: $$\frac{2}{3x - 4} = 7$$

Here’s a rewrite at a 12-year-old reading level, with clear steps and simple wording:

Since the variable is in the denominator, we need to remove it from there first.

The easiest way to do this is to multiply both sides of the equation by the entire denominator. This step helps cancel out the fraction.

Once the denominator is gone, the equation becomes simpler and easier to solve.

Right, they cancel each other out no matter how complicated the factor is. So when we multiply both sides of our equation by $3x - 4$, the denominator on the side with the rational expression gets cancelled out. $$\begin{align} \frac{2}{3x - 4} &= 7\\ \left(\frac{2}{3x - 4}\right)\color{#8925AE}{\times (3x-4)} &= 7\color{#8925AE}{\times (3x - 4)}\\ \frac{2\times (3x-4)}{3x - 4} &= 7\times 3x+7\times -4\\ 2 & = 21x -28 \end{align}$$

Above, we multiplied both sides of the equation by the rational expression's denominator. That made the denominator of the rational expression cancel out, just leaving us with $2$. On the other side of the equation, we used the distributive property to simplify the expression.

We can use addition to move the $28$ to the other side of the equation so that the $21x$ is just a little more on its own by just adding $28$ to both sides: $$\begin{align} 2 & = 21x -28\\\ 2 \color{#8925AE}{+28} &= 21x -28 \color{#8925AE}{+28}\\\ 30 &= 21x \end{align}$$

Using the inverse property, we know that $-28$ and $+28$ cancel each other out, so we've moved the number over.

To finish solving the rational expression, we just divide both sides by $21$ and then simplify: $$\begin{align} 30 &= 21x\\ \frac{30}{\color{#8925AE}{21}} &= \frac{21x}{\color{#8925AE}{21}}\\ \frac{30 \color{#8925AE}{\div 3}}{21 \color{#8925AE}{\div 3}} &= x\\ \frac{10}{7} &= x \end{align}$$

Dividing by $21$ means that the $21$ on the right of the equation canceled out, and then we removed the common factor ($3$) from the new fraction on the left, and we're done!

Looking Further

In later lessons, we will make common denominators for multiplying and dividing rational expressions.

(Video) Simplifying Rational Expressions

by YourMathGal

Here's a video by YourMathGal showing you how to simplify rational expressions. She goes through a couple problems showing you the step by step process.

$\large{\text{Summary}}$

To simplify a rational expression, factor out common terms in the numerator and denominator. Here's an example: $$\begin{align} \frac{5xy-3y}{9-15x}&=\frac{y(5x-3)}{3(3-5x)}\\ &=\frac{-y(3-5x)}{3(3-5x)}\\ &=\frac{-y\cancel{(3-5x)}}{3\cancel{(3-5x)}}\\ &=-\frac{y}{3} \end{align}$$ Be careful when dealing with addition and subtraction. You have to factor by the whole expression that contains the addition or subtraction. YourMathGal goes over a lot of these small deviations how how to solve them in the video.

$$\begin{align} \frac{x^2+8x}{88+11x}&=\frac{x(x+8)}{11(8+x)}\\ &=\frac{-x(8+x)}{11(8+x)}\\ &=\frac{-x\cancel{(8+x)}}{11\cancel{(8+x)}}\\ &=-\frac{x}{11} \end{align}$$

Intro to Rational Expressions and Simplifying

A rational expression is a fraction, but instead of just numbers, it has polynomials on the top and bottom. Rational expressions are often used to model real-world situations.

For example $\frac{x=2}{x+5}$ is a rational expression.

Just like regular fractions, we can sometimes simplify rational expressions. To do this, we look for common factors in the numerator and denominator.

If the same factor appears on both the top and bottom, we can cancel it out. We can do this because any number divided by itself equals $1$.

Factoring and simplifying can reveal removable discontinuities.

Review the rational function in the image above.

Let's look at another example.

####Example

Let's say we have $\frac{x^2+3x}{x}$.

First, we factor the top: $x^2+3x = x(x+3)$. Now, the rational expression looks like:

$$\frac{x(x+3)}{x}$$

We can cancel out the $x$ on the top and bottom because they're the same. That just leaves us with $x+3$.

Always remember: only cancel factors, not terms that are added or subtracted.

Looking Further

Now that we've covered the basics, we can continue to build on this idea. In the next few lessons, you will learn how to:

• Add and subtract rational expressions
• Multiply rational expressions
• Divide rational expressions